Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 943, September 2008  Screen Reader
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Abstract:
This paper assesses the empirical merits of PcGets and Autometricstwo recent algorithms for computerautomated model selectionusing them to improve upon Kamin and Ericsson's (1993) model of Argentine broad money demand. The selected model is an economically sensible and statistically satisfactory error correction model, in which cointegration between money, inflation, the interest rate, and exchange rate depreciation depends on the inclusion of a "ratchet" variable that captures irreversible effects of inflation. Shortrun dynamics differ markedly from the long run. Algorithmically based model selection complements opportunities for the researcher to contribute value added in the empirical analysis.
Keywords: Argentina, autometrics, broad money, dynamic specification, cointegration, conditional models, currency substitution, dollarization, error correction, exogeneity, hyperinflation, irreversibility, model design, model selection, money demand, PcGets, ratchet effect
JEL classification: C52, E41
We are delighted to contribute to this Festschrift in honor of David F. Hendry. As discussed in Ericsson (2004), David has contributed to numerous areas of econometrics and economics, including:
We draw on David's contributions to the first six topics to assess and improve upon Kamin and Ericsson's (1993) model of Argentine broad money demand, focusing on model design and cointegration analysis. Recent developments by David and coauthors in computerautomated model selection help us obtain a more parsimonious, empirically constant, datacoherent, error correction model for broad money demand in Argentina. Cointegration between money, inflation, the interest rate, and exchange rate depreciation depends on the inclusion of a "ratchet" variable that captures irreversible effects of inflation.
To better understand money demand and currency substitution in a hyperinflationary economy, Kamin and Ericsson (1993) develop an empirical model of broad money (M3) in Argentina for monthly data over 19781993, a period including hyperinflation and a subsequent decline in inflation to a rate close to contemporary U.S. and European levels. Kamin and Ericsson's underlying economic theory is a standard money demand model, augmented by shortrun nonlinear dynamics and a ratchet effect from inflation. Their empirical model clarifies the relative importance of factors determining money demand and currency holdings. Also, the structure of broad money demand in Argentina does not appear to have changed over the 1980s and 1990s. Rather, the fall in demand during the late 1980s and into the 1990s is explained by changes in the determinants of money demand itself.
That said, the analysis in Kamin and Ericsson (1993) has three notable shortcomings. First, their cointegration analysis excludes a trend, which may have affected inferences. Second, in their single equation modeling of Argentine money demand, Kamin and Ericsson augment the data from the cointegration analysis with an impulse dummy (for a known asset freeze from the Plan Bonex) and an asymmetric term in price acceleration. While both variables are stationary in principle, their exclusion from the cointegration analysis could have affected the results obtained. Third, an alternative single equation model might have been obtained if a different model search path had been followed.
Following the approach in Ericsson (2008, Chapters 9 and 10), the current paper addresses these issues, as follows. Cointegration is reanalyzed, including the impulse dummy, the asymmetric inflation term, and a trend. The cointegrating vector in this expanded framework is similar to the one obtained by Kamin and Ericsson (1993). Path dependence in model selection is examined by using two model selection algorithms: David Hendry and HansMartin Krolzig's (2001) PcGets, and Jurgen Doornik and David Hendry's (2007) Autometrics. Kamin and Ericsson's (1993) analysis is robust to multipath searches by both algorithms; at the same time, Autometrics finds an even more parsimonious specification. The details of the model improvement highlight the strengths and the limitations of computerautomated model selection. Our approach thus illustrates new techniques, which shed light on existing results. And, reexamination of an existing dataset with new techniques is very much in the spirit of other work in this area, including Hendry and Mizon (1978), Engle and Hendry (1993), Doornik, Hendry, and Nielsen (1998), and Hendry (2006).
This paper is organized as follows. Section 2 briefly describes the economic theory and the data. Section 3 summarizes the cointegration analysis and error correction model for Argentine money demand in Kamin and Ericsson (1993). Section 4 reanalyzes the longrun properties of Argentine money demand on the expanded dataset. Section 5 then designs a single equation model of money demand, using the algorithms for computerautomated model selection in PcGets and Autometrics. Depending upon the modeling strategy, presearch testing, choice of required regressors, and representation and choice of the initial general model, PcGets and Autometrics obtain several distinctalbeit similarfinal models in their generaltospecific selection processes. Additional analysis of those models obtains a final specification that is similar tobut more parsimonious thanthe one in Kamin and Ericsson (1993). That final specification appears wellspecified with empirically constant coefficients; and its economic interpretation is straightforward. Section 6 concludes.
For expositional convenience, two conventions are adopted. First, "domestic" means Argentine. Second, Argentine currency is always denominated in pesos (the Argentine currency at the end of the sample) although historically other currencies were used.
This section first discusses the theory of money demand (Section 2.1) and then considers the data themselves (Section 2.2).
The standard theory of money demand posits:
(1) 
where is nominal money demanded, is the price level, is a scale variable, and (in bold) is a vector of returns on various assets. The function is increasing in , decreasing in those elements of associated with assets excluded from , and increasing in those elements of for assets included in .
Three assets for Argentine residents are considered: broad money (M3), domestic goods, and dollars. Their nominal returns are denoted , , and , where is the exchange rate (domestic/foreign), variables in lowercase are in logarithms, and is the difference operator. This choice of assets and returns seems reasonable. Relatively few peso instruments outside of M3 were held in significant quantities during most of the sample period. Also, the interest rate on dollar deposits was small and unvarying relative to , so it was excluded in calculating the return on dollardenominated assets.
Empirical models below employ (1) in its standard loglinear form, with two modifications. First, the scale variable is omitted, as in Cagan's (1956) money demand model for hyperinflationary economies. ^{1} Second, following Enzler, Johnson, and Paulus (1976), Simpson and Porter (1980), Piterman (1988), Melnick (1990), Ahumada (1992, and Uribe (1997) inter alia, the money demand equation includes a ratchet variable, which is the maximum inflation rate to date, denoted . Higher inflation rates may induce innovations to economize on the use of domestic money balances. Once inflation subsides, these innovations are unlikely to disappear immediately (if at all), leading to a longlived negative effect of inflation on money demand. Hence, may proxy for financial innovation, be it a shift toward dollar usage or toward other forms of economizing on domestic money holdings.
With these two modifications, equation (1) has the following form:
(2) 
Anticipated signs of coefficients are , , , and . Broad money is composed primarily of interestbearing deposits, so the interest rate should exert a positive effect on money demand. The coefficients on and should be negative: goods and dollars are alternatives to holding money. Because increases monotonically throughout the sample, a strictly negative implies irreversible reductions in money demand due to historically higher rates of inflation.
If , , and enter equation (2) only as
relative rates of return, then
, and
equation (2) can be rewritten as:
(3) 
Equation (3) links real money demand to two opportunity costs and the ratchet effect. This representation is particularly useful when interpreting empirical error correction models in the context of multiple markets influencing money demand.
This subsection describes the data available and considers some of their basic properties. The data are a broad measure of money (M3), as measured by all pesodenominated currency and domestic bank deposits (, millions of pesos); the domestic consumer price index (, 1968 = 1.00); the interest rate on domestic pesodenominated 30day fixedterm bank deposits (, fraction at a monthly rate); and the freemarket exchange rate (, in pesos per dollar). Also, is transformed to the variable [denoted ] to measure the differential effect of positive (rather than negative) accelerations in prices, as in Ahumada (1992). The variable is interpretable as allowing asymmetric shortrun effects of inflation, similar to allowing asymmetric longrun effects. All data are monthly and seasonally unadjusted, over January 1977January 1993. Allowing for lags and transformations, estimation is over February 1978January 1993 () unless otherwise noted. Two dummy variables are also used: , an impulse dummy for the beginning of the Plan Bonex (January 1990); and , the seasonal dummy. Kamin and Ericsson (1993, Appendix) provide further details on the data.
Figure 1a plots the logarithms of nominal money and prices ( and ), which are notable by spanning orders of magnitude. Sharp increases in both series are visible around 1985 and 1989. While is the variable of central interest in this study, its evolution is most easily understood in light of the various rates of return. Figure 1b plots the (monthly) inflation rate , along with the generated ratchet variable . Figure 1c plots and the interest rate , which move closely together, albeit with inflation being more volatile on a monthtomonth basis. Figure 1d graphs and the depreciation in the nominal exchange rate , which also move closely together, with exchange rate depreciation being highly volatile. That said, real ex post monthly returns are commonly in excess of (plusorminus) two per cent, in large part owing to the high variability in the inflation rate.
Figure 1: The logarithms of nominal money and prices (m and p), inflation Δp and maximal inflation Δp^{max}, R and Δp, and R and Δe.
Data for Figure 1
Date  Panel 1a: m  Panel 1a: p+2.3$  Panel 1b: Δp  Panel 1b: Δp^{max}  Panel 1c: R  Panel 1c: Δp  Panel 1d: R  Panel 1d: Δe 

19771  10.857155  10.961625      0.073000    0.073000   
19772  10.758755  10.884239  0.077387  0.077387  0.074200  0.077387  0.074200  0.021706 
19773  10.665355  10.812413  0.071826  0.077387  0.067500  0.071826  0.067500  0.122035 
19774  10.510381  10.750043  0.062370  0.077387  0.064900  0.062370  0.064900  0.031535 
19775  10.362285  10.686959  0.063084  0.077387  0.061800  0.063084  0.061800  0.051217 
19776  10.273848  10.615349  0.071610  0.077387  0.062100  0.071610  0.062100  0.021707 
19777  10.183742  10.544732  0.070618  0.077387  0.066500  0.070618  0.066500  0.052291 
19778  10.101167  10.437101  0.107631  0.107631  0.073000  0.107631  0.073000  0.051739 
19779  10.035613  10.358629  0.078472  0.107631  0.079800  0.078472  0.079800  0.079201 
197710  9.969053  10.240148  0.118482  0.118482  0.092800  0.118482  0.092800  0.081071 
197711  9.892544  10.151973  0.088175  0.118482  0.102100  0.088175  0.102100  0.087666 
197712  9.736444  10.082810  0.069163  0.118482  0.103200  0.069163  0.103200  0.072623 
19781  9.648976  9.957366  0.125444  0.125444  0.099300  0.125444  0.099300  0.059934 
19782  9.552247  9.896122  0.061244  0.125444  0.080400  0.061244  0.080400  0.072993 
19783  9.484175  9.805323  0.090800  0.125444  0.070000  0.090800  0.070000  0.058169 
19784  9.387805  9.700686  0.104637  0.125444  0.066800  0.104637  0.066800  0.051728 
19785  9.295144  9.617891  0.082795  0.125444  0.068700  0.082795  0.068700  0.036894 
19786  9.174142  9.555416  0.062475  0.125444  0.071800  0.062475  0.071800  0.015402 
19787  9.092109  9.491317  0.064098  0.125444  0.068800  0.064098  0.068800  0.012762 
19788  9.004701  9.416266  0.075051  0.125444  0.067600  0.075051  0.067600  0.028552 
19789  8.941602  9.353338  0.062929  0.125444  0.062400  0.062929  0.062400  0.044944 
197810  8.876114  9.261066  0.092272  0.125444  0.064400  0.092272  0.064400  0.042898 
197811  8.795786  9.176594  0.084472  0.125444  0.067300  0.084472  0.067300  0.050277 
197812  8.696307  9.089823  0.086770  0.125444  0.070000  0.086770  0.070000  0.063505 
19791  8.606243  8.969195  0.120628  0.125444  0.067800  0.120628  0.067800  0.026983 
19792  8.522809  8.898115  0.071081  0.125444  0.063600  0.071081  0.063600  0.054739 
19793  8.442039  8.822912  0.075202  0.125444  0.063600  0.075202  0.063600  0.043862 
19794  8.358780  8.755501  0.067412  0.125444  0.064200  0.067412  0.064200  0.042590 
19795  8.267602  8.688789  0.066712  0.125444  0.065000  0.066712  0.065000  0.047245 
19796  8.165586  8.596120  0.092669  0.125444  0.066700  0.092669  0.066700  0.042813 
19797  8.082428  8.526803  0.069317  0.125444  0.069800  0.069317  0.069800  0.040016 
19798  7.992849  8.418576  0.108226  0.125444  0.073000  0.108226  0.073000  0.037462 
19799  7.908298  8.352147  0.066429  0.125444  0.073800  0.066429  0.073800  0.035247 
197910  7.798524  8.309923  0.042225  0.125444  0.071700  0.042225  0.071700  0.071391 
197911  7.719123  8.259724  0.050198  0.125444  0.062100  0.050198  0.062100  0.045074 
197912  7.624806  8.215239  0.044485  0.125444  0.059200  0.044485  0.059200  0.011181 
19801  7.555929  8.145804  0.069435  0.125444  0.057600  0.069435  0.057600  0.026769 
19802  7.493682  8.093531  0.052273  0.125444  0.051700  0.052273  0.051700  0.028371 
19803  7.448851  8.037352  0.056179  0.125444  0.048400  0.056179  0.048400  0.025772 
19804  7.423711  7.977454  0.059898  0.125444  0.044700  0.059898  0.044700  0.021437 
19805  7.404086  7.921217  0.056237  0.125444  0.045400  0.056237  0.045400  0.020672 
19806  7.339939  7.865372  0.055845  0.125444  0.053400  0.055845  0.053400  0.019462 
19807  7.262675  7.820649  0.044723  0.125444  0.060200  0.044723  0.060200  0.016708 
19808  7.200075  7.786930  0.033719  0.125444  0.050000  0.033719  0.050000  0.014678 
19809  7.158090  7.742520  0.044410  0.125444  0.043300  0.044410  0.043300  0.012905 
198010  7.118279  7.669200  0.073320  0.125444  0.043100  0.073320  0.043100  0.010585 
198011  7.077157  7.623486  0.045714  0.125444  0.046200  0.045714  0.046200  0.010012 
198012  6.999202  7.586041  0.037446  0.125444  0.054300  0.037446  0.054300  0.010119 
19811  6.987589  7.538262  0.047779  0.125444  0.056300  0.047779  0.056300  0.015168 
19812  6.976816  7.497327  0.040934  0.125444  0.066300  0.040934  0.066300  0.108388 
19813  6.968674  7.439160  0.058167  0.125444  0.081200  0.058167  0.081200  0.039644 
19814  6.902320  7.363212  0.075949  0.125444  0.074800  0.075949  0.074800  0.280808 
19815  6.871076  7.290576  0.072635  0.125444  0.080200  0.072635  0.080200  0.042066 
19816  6.797861  7.200960  0.089616  0.125444  0.101500  0.089616  0.101500  0.474811 
19817  6.700253  7.103440  0.097520  0.125444  0.108200  0.097520  0.108200  0.266651 
19818  6.621937  7.027211  0.076228  0.125444  0.102700  0.076228  0.102700  0.127959 
19819  6.522663  6.958166  0.069045  0.125444  0.083700  0.069045  0.083700  0.027101 
198110  6.452535  6.901579  0.056587  0.125444  0.069500  0.056587  0.069500  0.065734 
198111  6.393071  6.831991  0.069587  0.125444  0.073900  0.069587  0.073900  0.279507 
198112  6.278992  6.747626  0.084365  0.125444  0.069400  0.084365  0.069400  0.025449 
19821  6.220360  6.634984  0.112643  0.125444  0.072600  0.112643  0.072600  0.110983 
19822  6.153410  6.583486  0.051498  0.125444  0.071300  0.051498  0.071300  0.009184 
19823  6.099371  6.537398  0.046087  0.125444  0.068500  0.046087  0.068500  0.088953 
19824  6.034084  6.496378  0.041020  0.125444  0.082000  0.041020  0.082000  0.420186 
19825  5.974305  6.466232  0.030146  0.125444  0.074000  0.030146  0.074000  0.223469 
19826  5.926861  6.390238  0.075994  0.125444  0.058600  0.075994  0.058600  0.280055 
19827  5.882427  6.239592  0.150646  0.150646  0.051100  0.150646  0.051100  0.529450 
19828  5.875913  6.102704  0.136888  0.150646  0.049800  0.136888  0.049800  0.134455 
19829  5.849319  5.944955  0.157750  0.157750  0.069800  0.157750  0.069800  0.078225 
198210  5.770586  5.825489  0.119466  0.157750  0.069900  0.119466  0.069900  0.056761 
198211  5.683845  5.718025  0.107464  0.157750  0.084800  0.107464  0.084800  0.178078 
198212  5.511313  5.617094  0.100932  0.157750  0.084900  0.100932  0.084900  0.055395 
19831  5.348622  5.468777  0.148317  0.157750  0.104900  0.148317  0.104900  0.065786 
19832  5.270488  5.346381  0.122396  0.157750  0.099900  0.122396  0.099900  0.090393 
19833  5.184310  5.239619  0.106762  0.157750  0.100000  0.106762  0.100000  0.137318 
19834  5.077400  5.141867  0.097753  0.157750  0.100000  0.097753  0.100000  0.079840 
19835  4.956389  5.055090  0.086777  0.157750  0.100000  0.086777  0.100000  0.001308 
19836  4.836573  4.908198  0.146891  0.157750  0.089000  0.146891  0.089000  0.077235 
19837  4.735985  4.790825  0.117373  0.157750  0.105000  0.117373  0.105000  0.232099 
19838  4.618722  4.631741  0.159084  0.159084  0.116000  0.159084  0.116000  0.257112 
19839  4.495025  4.438088  0.193652  0.193652  0.142000  0.193652  0.142000  0.268383 
198310  4.336541  4.281312  0.156777  0.193652  0.145000  0.156777  0.145000  0.158403 
198311  4.152186  4.105393  0.175918  0.193652  0.145000  0.175918  0.145000  0.073155 
198312  3.888824  3.942416  0.162977  0.193652  0.145000  0.162977  0.145000  0.037707 
19841  3.709658  3.824530  0.117886  0.193652  0.115000  0.117886  0.115000  0.221229 
19842  3.558588  3.667592  0.156938  0.193652  0.100000  0.156938  0.100000  0.275041 
19843  3.423682  3.483124  0.184467  0.193652  0.100000  0.184467  0.100000  0.211368 
19844  3.298880  3.313404  0.169721  0.193652  0.130000  0.169721  0.130000  0.087301 
19845  3.169924  3.155722  0.157681  0.193652  0.130000  0.157681  0.130000  0.172524 
19846  2.996246  2.990957  0.164766  0.193652  0.130000  0.164766  0.130000  0.065345 
19847  2.830250  2.823051  0.167906  0.193652  0.155000  0.167906  0.155000  0.086602 
19848  2.661351  2.617280  0.205770  0.205770  0.155000  0.205770  0.155000  0.262119 
19849  2.529654  2.373963  0.243317  0.243317  0.155000  0.243317  0.155000  0.142697 
198410  2.410457  2.197303  0.176660  0.243317  0.170000  0.176660  0.170000  0.076308 
198411  2.216226  2.057763  0.139539  0.243317  0.170000  0.139539  0.170000  0.311530 
198412  2.026495  1.878126  0.179637  0.243317  0.170000  0.179637  0.170000  0.079912 
19851  1.844120  1.653950  0.224176  0.243317  0.175000  0.224176  0.175000  0.283278 
19852  1.672341  1.466076  0.187874  0.243317  0.180000  0.187874  0.180000  0.279051 
19853  1.518627  1.231011  0.235065  0.243317  0.200000  0.235065  0.200000  0.239069 
19854  1.280792  0.972767  0.258244  0.258244  0.240000  0.258244  0.240000  0.269167 
19855  1.032818  0.748694  0.224073  0.258244  0.300000  0.224073  0.300000  0.160410 
19856  0.765922  0.482222  0.266472  0.266472  0.160000  0.266472  0.160000  0.253104 
19857  0.629167  0.422124  0.060098  0.266472  0.035000  0.060098  0.035000  0.167337 
19858  0.560295  0.391946  0.030178  0.266472  0.035000  0.030178  0.035000  0.009836 
19859  0.486941  0.372189  0.019757  0.266472  0.035000  0.019757  0.035000  0.013240 
198510  0.405698  0.352912  0.019277  0.266472  0.031000  0.019277  0.031000  0.016050 
198511  0.372643  0.329503  0.023409  0.266472  0.031000  0.023409  0.031000  0.029143 
198512  0.252200  0.298290  0.031213  0.266472  0.031000  0.031213  0.031000  0.048801 
19861  0.204695  0.268444  0.029846  0.266472  0.031000  0.029846  0.031000  0.050383 
19862  0.146945  0.251688  0.016757  0.266472  0.031000  0.016757  0.031000  0.043814 
19863  0.112600  0.206273  0.045415  0.266472  0.031000  0.045415  0.031000  0.054048 
19864  0.060502  0.160015  0.046258  0.266472  0.031000  0.046258  0.031000  0.014350 
19865  0.001138  0.120536  0.039479  0.266472  0.031000  0.039479  0.031000  0.023821 
19866  0.067780  0.076084  0.044451  0.266472  0.033000  0.044451  0.033000  0.005586 
19867  0.130938  0.010650  0.065435  0.266472  0.035000  0.065435  0.035000  0.021883 
19868  0.162822  0.073552  0.084201  0.266472  0.051000  0.084201  0.051000  0.171870 
19869  0.209891  0.143366  0.069815  0.266472  0.045000  0.069815  0.045000  0.117772 
198610  0.323973  0.202126  0.058759  0.266472  0.050000  0.058759  0.050000  0.019841 
198611  0.386451  0.253722  0.051597  0.266472  0.055000  0.051597  0.055000  0.119049 
198612  0.521417  0.300035  0.046312  0.266472  0.055000  0.046312  0.055000  0.147807 
19871  0.578201  0.372909  0.072874  0.266472  0.055000  0.072874  0.055000  0.090841 
19872  0.609968  0.435909  0.063000  0.266472  0.060000  0.063000  0.060000  0.000776 
19873  0.682155  0.514714  0.078805  0.266472  0.030000  0.078805  0.030000  0.093407 
19874  0.731072  0.547805  0.033092  0.266472  0.042000  0.033092  0.042000  0.081494 
19875  0.794314  0.588707  0.040901  0.266472  0.047000  0.040901  0.047000  0.013330 
19876  0.875165  0.665680  0.076974  0.266472  0.065000  0.076974  0.065000  0.004614 
19877  0.956445  0.762078  0.096398  0.266472  0.075000  0.096398  0.075000  0.138354 
19878  1.009318  0.890617  0.128539  0.266472  0.095000  0.128539  0.095000  0.204818 
19879  1.098891  1.001141  0.110524  0.266472  0.110000  0.110524  0.110000  0.164973 
198710  1.217278  1.179679  0.178538  0.266472  0.135000  0.178538  0.135000  0.136400 
198711  1.326410  1.277429  0.097750  0.266472  0.089000  0.097750  0.089000  0.028094 
198712  1.460809  1.310883  0.033454  0.266472  0.124000  0.033454  0.124000  0.116676 
19881  1.534796  1.397910  0.087027  0.266472  0.132000  0.087027  0.132000  0.176399 
19882  1.622239  1.497143  0.099232  0.266472  0.133000  0.099232  0.133000  0.057968 
19883  1.770242  1.634633  0.137490  0.266472  0.156000  0.137490  0.156000  0.090983 
19884  1.879811  1.793618  0.158984  0.266472  0.162000  0.158984  0.162000  0.090271 
19885  2.016426  1.939633  0.146015  0.266472  0.173000  0.146015  0.173000  0.117629 
19886  2.198877  2.104847  0.165215  0.266472  0.195000  0.165215  0.195000  0.273394 
19887  2.379716  2.333106  0.228259  0.266472  0.227000  0.228259  0.227000  0.172848 
19888  2.595137  2.577033  0.243927  0.266472  0.108000  0.243927  0.108000  0.147626 
19889  2.748841  2.687613  0.110580  0.266472  0.091000  0.110580  0.091000  0.013891 
198810  2.840920  2.773728  0.086115  0.266472  0.093000  0.086115  0.093000  0.043619 
198811  2.966466  2.829298  0.055570  0.266472  0.102000  0.055570  0.102000  0.029556 
198812  3.160743  2.895457  0.066159  0.266472  0.122000  0.066159  0.122000  0.021767 
19891  3.329408  2.980918  0.085461  0.266472  0.121000  0.085461  0.121000  0.066074 
19892  3.415708  3.072503  0.091586  0.266472  0.190000  0.091586  0.190000  0.401292 
19893  3.538077  3.229550  0.157047  0.266472  0.216000  0.157047  0.216000  0.477542 
19894  3.720073  3.517527  0.287976  0.287976  0.447000  0.287976  0.447000  0.461351 
19895  4.078186  4.096765  0.579239  0.579239  1.154000  0.579239  1.154000  0.689090 
19896  4.758953  4.859782  0.763017  0.763017  1.369000  0.763017  1.369000  1.143784 
19897  5.474110  5.947093  1.087311  1.087311  0.339000  1.087311  0.339000  0.473116 
19898  6.023763  6.268176  0.321083  1.087311  0.128000  0.321083  0.128000  0.033621 
19899  6.250978  6.357611  0.089435  1.087311  0.074000  0.089435  0.074000  0.023767 
198910  6.420034  6.412053  0.054442  1.087311  0.061000  0.054442  0.061000  0.074776 
198911  6.496225  6.475218  0.063164  1.087311  0.096000  0.063164  0.096000  0.229980 
198912  6.546786  6.812210  0.336993  1.087311  0.553000  0.336993  0.553000  0.399647 
19901  6.413787  7.395565  0.583354  1.087311  0.264000  0.583354  0.264000  0.243769 
19902  6.698515  7.875335  0.479771  1.087311  0.361000  0.479771  0.361000  0.760012 
19903  6.988321  8.545857  0.670521  1.087311  0.456000  0.670521  0.456000  0.309943 
19904  7.417761  8.653568  0.107711  1.087311  0.116000  0.107711  0.116000  0.022064 
19905  7.681653  8.781147  0.127579  1.087311  0.088000  0.127579  0.088000  0.000000 
19906  7.882654  8.911279  0.130132  1.087311  0.140000  0.130132  0.140000  0.050644 
19907  8.064762  9.014065  0.102786  1.087311  0.111000  0.102786  0.111000  0.033617 
19908  8.155649  9.156759  0.142694  1.087311  0.098000  0.142694  0.098000  0.129045 
19909  8.239013  9.302393  0.145635  1.087311  0.167000  0.145635  0.167000  0.099192 
199010  8.361124  9.376484  0.074091  1.087311  0.109000  0.074091  0.109000  0.008054 
199011  8.479201  9.436455  0.059971  1.087311  0.067000  0.059971  0.067000  0.082366 
199012  8.630933  9.482158  0.045702  1.087311  0.067000  0.045702  0.067000  0.085954 
19911  8.709647  9.556325  0.074167  1.087311  0.135900  0.074167  0.135900  0.522751 
19912  8.769492  9.795266  0.238941  1.087311  0.167900  0.238941  0.167900  0.057748 
19913  8.869370  9.900007  0.104741  1.087311  0.115900  0.104741  0.115900  0.036219 
19914  8.990641  9.953648  0.053641  1.087311  0.014200  0.053641  0.014200  0.018529 
19915  9.066966  9.981307  0.027659  1.087311  0.015500  0.027659  0.015500  0.010147 
19916  9.125882  10.012066  0.030760  1.087311  0.017000  0.030760  0.017000  0.006541 
19917  9.164841  10.037646  0.025580  1.087311  0.018000  0.025580  0.018000  0.001506 
19918  9.211370  10.050571  0.012925  1.087311  0.014000  0.012925  0.014000  0.000502 
19919  9.255362  10.068411  0.017840  1.087311  0.011000  0.017840  0.011000  0.005537 
199110  9.306332  10.082314  0.013903  1.087311  0.011000  0.013903  0.011000  0.000000 
199111  9.362787  10.086306  0.003992  1.087311  0.011000  0.003992  0.011000  0.000000 
199112  9.451363  10.092288  0.005982  1.087311  0.013000  0.005982  0.013000  0.008044 
19921  9.502009  10.121825  0.029537  1.087311  0.011000  0.029537  0.011000  0.008044 
19922  9.529100  10.143137  0.021312  1.087311  0.010000  0.021312  0.010000  0.000000 
19923  9.545197  10.163915  0.020779  1.087311  0.009000  0.020779  0.009000  0.002017 
19924  9.591533  10.176701  0.012785  1.087311  0.010000  0.012785  0.010000  0.002017 
19925  9.670212  10.183409  0.006708  1.087311  0.009000  0.006708  0.009000  0.000000 
19926  9.740716  10.191219  0.007810  1.087311  0.008000  0.007810  0.008000  0.000000 
19927  9.789311  10.208359  0.017140  1.087311  0.010000  0.017140  0.010000  0.000000 
19928  9.820106  10.223214  0.014855  1.087311  0.009000  0.014855  0.009000  0.000000 
19929  9.846282  10.233499  0.010285  1.087311  0.009000  0.010285  0.009000  0.000000 
199210  9.858072  10.246079  0.012580  1.087311  0.009000  0.012580  0.009000  0.000000 
199211  9.870603  10.250678  0.004599  1.087311  0.010000  0.004599  0.010000  0.000000 
199212  9.900182  10.253511  0.002833  1.087311  0.011000  0.002833  0.011000  0.000000 
19931  9.949273  10.261800  0.008289  1.087311  0.009000  0.008289  0.009000  0.000505 
19932    10.269078  0.007278  1.087311  0.010500  0.007278  0.010500   
19933        1.087311         
19934        1.087311         
19935        1.087311         
19936        1.087311         
19937        1.087311         
19938        1.087311         
19939        1.087311         
199310        1.087311         
199311        1.087311         
199312        1.087311         
Figure 2: The logarithms of real money (m  p), and the negative of the maximal inflation rate (Δp^{max}), adjusted for means.
Data for Figure 2
Date  (m  p)  (Δp^{max} + 2.7) 

19771  2.4044708   
19772  2.4254841  2.6226133 
19773  2.4470578  2.6226133 
19774  2.5396620  2.6226133 
19775  2.6246740  2.6226133 
19776  2.6415009  2.6226133 
19777  2.6609896  2.6226133 
19778  2.6359337  2.5923693 
19779  2.6230167  2.5923693 
197710  2.5710950  2.5815184 
197711  2.5594291  2.5815184 
197712  2.6463657  2.5815184 
19781  2.6083901  2.5745561 
19782  2.6438750  2.5745561 
19783  2.6211474  2.5745561 
19784  2.6128812  2.5745561 
19785  2.6227467  2.5745561 
19786  2.6812740  2.5745561 
19787  2.6992086  2.5745561 
19788  2.7115654  2.5745561 
19789  2.7117352  2.5745561 
197810  2.6849520  2.5745561 
197811  2.6808076  2.5745561 
197812  2.6935160  2.5745561 
19791  2.6629523  2.5745561 
19792  2.6753058  2.5745561 
19793  2.6808731  2.5745561 
19794  2.6967204  2.5745561 
19795  2.7211864  2.5745561 
19796  2.7305335  2.5745561 
19797  2.7443747  2.5745561 
19798  2.7257271  2.5745561 
19799  2.7438499  2.5745561 
197910  2.8113982  2.5745561 
197911  2.8406013  2.5745561 
197912  2.8904332  2.5745561 
19801  2.8898747  2.5745561 
19802  2.8998490  2.5745561 
19803  2.8885010  2.5745561 
19804  2.8537433  2.5745561 
19805  2.8171301  2.5745561 
19806  2.8254328  2.5745561 
19807  2.8579744  2.5745561 
19808  2.8868550  2.5745561 
19809  2.8844296  2.5745561 
198010  2.8509213  2.5745561 
198011  2.8463295  2.5745561 
198012  2.8868387  2.5745561 
19811  2.8506729  2.5745561 
19812  2.8205113  2.5745561 
19813  2.7704866  2.5745561 
19814  2.7608915  2.5745561 
19815  2.7195002  2.5745561 
19816  2.7030990  2.5745561 
19817  2.7031864  2.5745561 
19818  2.7052739  2.5745561 
19819  2.7355030  2.5745561 
198110  2.7490435  2.5745561 
198111  2.7389203  2.5745561 
198112  2.7686345  2.5745561 
19821  2.7146243  2.5745561 
19822  2.7300761  2.5745561 
19823  2.7380279  2.5745561 
19824  2.7622941  2.5745561 
19825  2.7919270  2.5745561 
19826  2.7633769  2.5745561 
19827  2.6571650  2.5493543 
19828  2.5267911  2.5493543 
19829  2.3956354  2.5422503 
198210  2.3549030  2.5422503 
198211  2.3341807  2.5422503 
198212  2.4057812  2.5422503 
19831  2.4201552  2.5422503 
19832  2.3758929  2.5422503 
19833  2.3553089  2.5422503 
19834  2.3644663  2.5422503 
19835  2.3987012  2.5422503 
19836  2.3716257  2.5422503 
19837  2.3548398  2.5422503 
19838  2.3130190  2.5409155 
19839  2.2430633  2.5063478 
198310  2.2447708  2.5063478 
198311  2.2532078  2.5063478 
198312  2.3535920  2.5063478 
19841  2.4148719  2.5063478 
19842  2.4090035  2.5063478 
19843  2.3594421  2.5063478 
19844  2.3145235  2.5063478 
19845  2.2857986  2.5063478 
19846  2.2947101  2.5063478 
19847  2.2928006  2.5063478 
19848  2.2559294  2.4942296 
19849  2.1443092  2.4566829 
198410  2.0868463  2.4566829 
198411  2.1415376  2.4566829 
198412  2.1516308  2.4566829 
19851  2.1098295  2.4566829 
19852  2.0937349  2.4566829 
19853  2.0123842  2.4566829 
19854  1.9919753  2.4417561 
19855  2.0158762  2.4417561 
19856  2.0162998  2.4335278 
19857  2.0929571  2.4335278 
19858  2.1316508  2.4335278 
19859  2.1852480  2.4335278 
198510  2.2472136  2.4335278 
198511  2.2568599  2.4335278 
198512  2.3460897  2.4335278 
19861  2.3637493  2.4335278 
19862  2.4047426  2.4335278 
19863  2.3936728  2.4335278 
19864  2.3995127  2.4335278 
19865  2.4216740  2.4335278 
19866  2.4438638  2.4335278 
19867  2.4415871  2.4335278 
19868  2.3892707  2.4335278 
19869  2.3665251  2.4335278 
198610  2.4218475  2.4335278 
198611  2.4327286  2.4335278 
198612  2.5213821  2.4335278 
19871  2.5052922  2.4335278 
19872  2.4740590  2.4335278 
19873  2.4674419  2.4335278 
19874  2.4832662  2.4335278 
19875  2.5056079  2.4335278 
19876  2.5094849  2.4335278 
19877  2.4943674  2.4335278 
19878  2.4187012  2.4335278 
19879  2.3977493  2.4335278 
198710  2.3375993  2.4335278 
198711  2.3489815  2.4335278 
198712  2.4499263  2.4335278 
19881  2.4368860  2.4335278 
19882  2.4250962  2.4335278 
19883  2.4356083  2.4335278 
19884  2.3861927  2.4335278 
19885  2.3767931  2.4335278 
19886  2.3940302  2.4335278 
19887  2.3466101  2.4335278 
19888  2.3181037  2.4335278 
19889  2.3612281  2.4335278 
198810  2.3671925  2.4335278 
198811  2.4371680  2.4335278 
198812  2.5652860  2.4335278 
19891  2.6484900  2.4335278 
19892  2.6432048  2.4335278 
19893  2.6085265  2.4335278 
19894  2.5025463  2.4120239 
19895  2.2814211  2.1207613 
19896  2.1991703  1.9369831 
19897  1.8270167  1.6126887 
19898  2.0555861  1.6126887 
19899  2.1933668  1.6126887 
198910  2.3079804  1.6126887 
198911  2.3210075  1.6126887 
198912  2.0345762  1.6126887 
19901  1.3182223  1.6126887 
19902  1.1231794  1.6126887 
19903  0.7424642  1.6126887 
19904  1.0641926  1.6126887 
19905  1.2005059  1.6126887 
19906  1.2713758  1.6126887 
19907  1.3506973  1.6126887 
19908  1.2988905  1.6126887 
19909  1.2366191  1.6126887 
199010  1.2846400  1.6126887 
199011  1.3427452  1.6126887 
199012  1.4487749  1.6126887 
19911  1.4533218  1.6126887 
19912  1.2742258  1.6126887 
19913  1.2693630  1.6126887 
19914  1.3369932  1.6126887 
19915  1.3856598  1.6126887 
19916  1.4138158  1.6126887 
19917  1.4271944  1.6126887 
19918  1.4607986  1.6126887 
19919  1.4869504  1.6126887 
199110  1.5240181  1.6126887 
199111  1.5764808  1.6126887 
199112  1.6590749  1.6126887 
19921  1.6801847  1.6126887 
19922  1.6859629  1.6126887 
19923  1.6812817  1.6126887 
19924  1.7148326  1.6126887 
19925  1.7868029  1.6126887 
19926  1.8494967  1.6126887 
19927  1.8809513  1.6126887 
19928  1.8968918  1.6126887 
19929  1.9127827  1.6126887 
199210  1.9119928  1.6126887 
199211  1.9199251  1.6126887 
199212  1.9466715  1.6126887 
19931  1.9874732  1.6126887 
19932    1.6126887 
19933    1.6126887 
19934    1.6126887 
19935    1.6126887 
19936    1.6126887 
19937    1.6126887 
19938    1.6126887 
19939    1.6126887 
199310    1.6126887 
199311    1.6126887 
199312    1.6126887 
The overall behavior of inflation (and so of and ) can be characterized by periods of increasing inflation, followed by government "plans" to reign in inflation. The acceleration of prices during the early 1980s was sharply reversed in mid1985 by the Plan Austral, which combined wage, price, and exchange rate freezes with some fiscal adjustment. Reductions in the fiscal deficit were not sufficient to eliminate inflationary pressures, which resumed in earnest by 1987. The August 1988 Plan Primavera ("Spring Plan") followed, and it aimed to limit the growth of prices and the official exchange rate to 4 percent per month. While inflation fell temporarily, the real exchange rate appreciated and the fiscal situation deteriorated. In February 1989, the Central Bank floated the exchange rate for financial transactions, which promptly depreciated sharply; and inflation rapidly increased to a record 197 percent per month in July 1989.
Under newly elected President Menem, the authorities announced a new program similar to the Plan Austral. Initially, inflation fell dramatically; but appreciation of the real exchange rate forced the Central Bank to float the commercial exchange rate, which quickly depreciated in value and spurred price inflation. In January 1990, the authorities attempted to restrain inflation by freezing most domestic pesodenominated bank time deposits and converting them to 10year dollardenominated bonds known as Bonex. The socalled Plan Bonex had little immediate effect upon inflation, but it did further reduce the Argentine public's faith in their financial system. By March 1990, when inflation reached 95.5 percent per month, broad money reached a record low of 3.1 percent of GDP .
Subsequently, inflation declined to singledigit levels due to a reduction in monetary emission made possible by concerted efforts to achieve fiscal adjustment. The fiscal deficit declined from over 20 percent of GDP in 1989 to about 3 percent in 1990 and 2 percent in 1991. In March 1991, the government announced the "Convertibility Program," which fixed the exchange rate against the dollar and required the Central Bank to hold international reserves equivalent to the monetary base. Subsequently, the inflation rate fell to under 1 percent per month.
Figure 2 graphs the log of real money () and the negative of the ratchet variable . Real money initially increases gradually, then falls abruptly by 20% in 1982. After continuing to fall through 1984, real money increases until the hyperinflation in 1989, when it plummets to approximately half its "prehyper" level. Even after very low inflation in subsequent years, real money did not return to its level of early 1989. Declines in real money are closely correlated with increases in the ratchet variable, although the stability of a relation between these variables may be an issue, noting the remaining large deviations between them.^{2}
This section summarizes the model of Argentine money demand developed by Kamin and Ericsson (1993).
Kamin and Ericsson (1993) test for and find cointegration between real money, the interest rate, the inflation rate, exchange rate depreciation, and the ratchet variable; and the ratchet variable is key to finding cointegration. While the interest rate and the exchange rate do not appear to be weakly exogenous, there are only minor differences between system estimates of the cointegrating vector and the solved longrun coefficients from a conditional singleequation autoregressive distributed lag (ADL) model. So, Kamin and Ericsson (1993) model broad money as a singleequation conditional error correction model (ECM).
In their single equation modeling, Kamin and Ericsson (1993) start with a seventhorder ADL that has 63 coefficients and simplify it to a more restricted "intermediate" ADL with only 30 coefficients. Kamin and Ericsson (1993) then further simplify to obtain the following 16coefficient model, which is their equation (6).
Δ  (4)  
where a circumflex on the dependent variable denotes its fitted value, the subscript is the time index, , R^{2} is the squared multiple correlation coefficient, and is the estimated residual standard error. The longrun solution to equation (4) is:
(5) 
Kamin and Ericsson (1993) show that equation (4) has a straightforward economic interpretation and is statistically satisfactory. Economically, the longrun coefficients in (5) satisfy sign restrictions that are consonant with a money demand function. The shortrun variables and coefficients in (4) are also easily understood. Each shortrun variable enters as a second difference (an acceleration), which is a natural transformation to stationarity for a potentially I(2) variable. The coefficient on is close to 1, implying that, in the short run, agents are in essence adjusting nominal (and not real) money.^{3} The lag lengths on , , and are consistent with agents' adjustments for seasonality in the data. The variable is also consistent with a natural databased predictor of future (seasonal) inflation, extending the theoretical and empirical developments on such predictors in Flemming (1976), Hendry and Ericsson (1991), and Campos and Ericsson (1999). And, the coefficient on is very negative and statistically significant, implying stronger reactions to rising inflation than to falling inflation.
The estimated money demand function also shed lights on the dollarization of the Argentine economy. Kamin and Ericsson (2003) reinterpret the ratchet effect in light of data measuring the extent of dollarization. Specifically, the reduction in peso money demand attributable to the ratchet effect is comparable in magnitude to the estimated stock of total dollar assets held domestically by Argentine residents, where those assets are estimated from U.S. Treasury data. This suggests that secular reductions in the demand for pesos reflect substitution into dollars rather than mere economizing on peso balances (or other forms of financial innovation). Thus, the ratchet may proxy for dollar holdings, which relaxes the draconian assumption of true irreversibility.
Statistically, Kamin and Ericsson (1993) show that equation (4) is parsimonious and empirically constant and satisfies a variety of diagnostic tests. Equation (4) and the regressions below report diagnostic statistics for testing against various alternative hypotheses: residual autocorrelation ( and ), skewness and excess kurtosis (), autoregressive conditional heteroscedasticity (), RESET (), heteroscedasticity ( and ), noninnovation errors relative to a more general model (), and predictive failure (, Chow's prediction interval statistic). The asymptotic null distribution is designated by or , where the degrees of freedom fill the parentheses. Estimated standard errors are in parentheses , below coefficient estimates; heteroscedasticityconsistent standard errors are in brackets . See Doornik and Hendry (2007) for details and references.
In spite of the apparent robustness of equation (4), its design has shortcomings. The associated cointegration analysis excludes , a linear trend, and an impulse dummy for the Plan Bonex. And, equation (4) may depend on the path taken for model selection. The remainder of the current paper addresses these issues.
This section presents unit root tests for the variables of interest (Section 4.1). Then, Johansen's maximum likelihood procedure is applied to test for cointegration among real money, inflation, the interest rate, exchange rate depreciation, , the ratchet variable, and a linear trend (Section 4.2). Coefficient restrictions and the adjustment mechanism are examined in the Johansen framework.
Table 1 lists augmented DickeyFuller (ADF) statistics and related calculations for the data. In order to test whether a given series is I(0), I(1), I(2), or I(3), Table 1 calculates unit root tests for the original variables, for their changes, and for the changes of the changes. This permits testing the order of integration, albeit by testing adjacent orders of integration in a pairwise fashion. The largest estimated root () is listed adjacent to each ADF statistic: this root should be approximately unity if the null hypothesis is correct. The lag length of the reported ADF regression is based on minimizing the AIC, starting with a maximum of twelve lags.
Table 1: ADF statistics for testing a unit root in various time series
Variable^{a,b}  lag  tprob (%)  Fprob (%)  AIC  

m  8  2.81  0.988  5.318  2.1  52.1  5.75 
p  12  2.95  0.984  7.741  17.8    4.98 
e  8  3.07  0.970  12.63  0.1  43.7  4.02 
m  p  11  3.08  0.934  6.912  2.9  47.1  5.22 
R  12  2.06  0.821  9.208  8.2    4.64 
Δp^{max}  10  1.74  0.983  2.631  5.7  62.9  7.15 
R  Δp  5  4.64**  0.079  9.300  0.1  12.4  4.65 
R  Δe  1  9.72**  0.069  10.65  1.4  24.2  4.40 
Δm  8  2.28  0.849  5.431  15.8  48.6  5.71 
Δp  12  2.45  0.810  7.905  8.1    4.94 
Δe  8  2.76  0.703  12.96  13.7  84.6  3.97 
Δ(m  p)  7  3.95*  0.380  7.118  14.0  67.5  5.18 
ΔR  11  5.46**  1.952  9.334  2.4  43.5  4.61 
Δ(Δp^{max})  9  4.27**  0.487  2.657  3.0  76.4  7.14 
Δ(R Δp)  11  6.16**  5.268  9.670  1.1  81.7  4.54 
Δ(R Δe)  12  7.49**  7.758  10.71  9.1    4.33 
Δ^{2}m  6  9.12**  1.348  5.520  1.7  40.8  5.69 
Δ^{2}p  10  4.96**  1.517  8.072  11.7  43.9  4.91 
Δ^{2}e  6  9.36**  2.257  13.26  0.0  66.7  3.94 
Δ^{2}(m  p)  9  7.56**  3.759  7.323  1.2  46.0  5.11 
Δ^{2}R  12  7.87**  11.81  9.888  4.1    4.49 
Δ^{2}(Δp^{max})  10  6.09**  2.108  2.778  11.6  78.5  7.04 
Δ^{2}(R  Δp)  11  7.67**  17.68  10.65  2.3  86.0  4.35 
Δ^{2}(R  Δe)  12  8.49**  19.20  12.19  0.2    4.07 
Δ^{2}p^{pos}  5  3.17  0.643  4.188  0.7  76.7  6.24 
Notes:
a. Twelfthorder ADF regressions were
initially estimated, and the final lag length was selected to
minimize the Akaike Information Criterion (AIC). The columns report
the name of the variable examined, the selected lag
length , the ADF statistic on the
simplified regression (), the estimated coefficient on
the lagged level that is being tested for a unit value (), the regression's residual
standard error (, in %), the tail
probability of the tstatistic on the longest
lag of the final regression (tprob, in %),
the tail probability of the Fstatistic for the
lags dropped (Fprob, in %), and the
AIC.
b. All of the ADF regressions include an intercept, monthly
dummies, and a linear trend. MacKinnon's (1991) approximate
finitesample critical values for the corresponding ADF statistics
are 3.14 (10%), 3.44 (5%), and 4.01 (1%) for T = 177. In this table, and in the other
results reported herein, rejection of the indicated null hypothesis
is denoted by ^{+}, *, and ** for the 10%, 5%, and 1%
levels. Samples sizes are T = 179, T = 178, and T = 177 respectively for the three
null hypotheses.
Nominal money, prices, and the exchange rate appear to be I(2). Real money, the nominal interest rate, inflation, and the inflation ratchet variable appear to be I(1). The ex post real interest rate and appear stationary.
Cointegration analysis helps clarify the longrun relationships between integrated variables. A brief review leads to the current analysis and places the latter in context.
Johansen's (1988, 1991) procedure is maximum likelihood for finiteorder vector autoregressions (VARs) with variables that are integrated of order one [I(1)], and it is easily calculated for such systems. Various approaches exist for modeling possibly cointegrated I(2) variables. Johansen (1992b) proposes and implements a unified (vector autoregressive) system approach for the entire testing sequence going from I(2) to I(1) to I(0). His empirical application uses data on U.K. narrow money demand, which appear to have the same orders of integration as the Argentine series above. For the U.K. data, Johansen (1992b) tests for and finds that nominal money and prices (which are I(2)) cointegrate with a (+1 : 1) cointegrating vector to give real money, which is I(1). He then tests for and finds that real money, inflation, real income, and interest rates (all of which are I(1)) cointegrate. Because the I(2) Argentine variables and appear to cointegrate as the I(1) variable , the cointegration analysis here begins with the variables , , , , , , and a linear trend.
Empirically, the lag order of the VAR is not known a priori, so some testing of lag order may be fruitful in order to ensure reasonable power of the Johansen procedure. Given the number of variables, the number of observations, and the data's periodicity, the largest system considered is a seventhorder VAR of , , , , , and . In that VAR, the linear trend is restricted to lie in the cointegration space; and an intercept, seasonal dummies, and the Plan Bonex dummy (and three of its lags) enter freely. Empirically, the seventh lag may be statistically insignificant, but no further lag restrictions appear feasible, so inferences below are for the seventhorder VAR .
Table 2 reports the standard statistics, 95% critical values (c.v.'s), and estimates for Johansen's procedure applied to this seventhorder VAR . The maximal eigenvalue and trace eigenvalue statistics ( and ) strongly reject the null of no cointegration in favor of at least one cointegrating relationship, and likely in favor of two cointegrating relationships. However, parallel statistics with a degreesoffreedom adjustment ( and ) suggest only one cointegrating relationship. Because the VAR for Table 2 uses a large number of degrees of freedom in estimation, inferences are based on the adjusted eigenvalue statistics.
Table 2: A cointegration analysis of the Argentine money demand data: Panel A: Rank r
Rank r  r = 0  r ≤ 1  r ≤ 2  r ≤ 3  r ≤ 4  r ≤ 5  r ≤ 6 

Loglikelihood  2497.21  2528.69  2551.50  2563.38  2571.45  2576.82  2578.52 
Eigenvalue λ_{r}    0.295  0.224  0.124  0.086  0.058  0.019 
Table 2: A cointegration analysis of the Argentine money demand data: Panel B: Null hypothesis
Statistic  r = 0  r ≤ 1  r ≤ 2  r ≤ 3  r ≤ 4  r ≤ 5 

λ_{max}  62.98**  45.61**  23.77  16.13  10.75  3.38 
λ^{a}_{max}  48.28*  34.97  18.22  12.37  8.24  2.59 
95% c.v.  43.97  37.52  31.46  25.54  18.96  12.25 
λ_{trace}  162.6**  99.64**  54.03  30.26  14.13  3.38 
λ^{a}_{trace}  124.7*  76.39  41.42  23.20  10.83  2.59 
95% c.v.  114.9  87.31  62.99  42.44  25.32  12.25 
Table 2: A cointegration analysis of the Argentine money demand data: Panel C: Eigenvectors β'
Variable  (m  p) 
Δp 
R 
Δp^{max} 
Δe 
Δ^{2}p^{pos} 
trend 

Row 1  1 
10.89 
17.53 
1.20 
6.17 
62.69 
0.0028 
Row 2  0.08 
1 
0.78 
0.04 
0.30 
0.50 
0.0003 
Row 3  0.25 
2.40 
1 
0.29 
0.49 
4.66 
0.0002 
Row 4  0.61 
8.49 
47.66 
1 
17.90 
5.39 
0.0062 
Row 5  1.43 
15.50 
18.97 
0.38 
1 
7.94 
0.0092 
Row 6  0.63 
0.34 
2.35 
1.27 
0.35 
1 
0.0058 
Table 2: A cointegration analysis of the Argentine money demand data: Panel D: Adjustment Coefficients α
Variable  Column 1  Column 2  Column 3  Column 4  Column 5  Column 6 

(m  p)  0.020  0.265  0.083  0.002  0.010  0.010 
Δp  0.015  0.365  0.015  0.002  0.013  0.002 
R  0.034  0.085  0.067  0.002  0.020  0.015 
Δp^{max}  0.016  0.137  0.005  0.002  0.004  0.002 
Δe  0.048  0.877  0.093  0.011  0.034  0.012 
Δ^{2}p^{pos}  0.005  0.344  0.052  0.000  0.009  0.001 
Table 2: A cointegration analysis of the Argentine money demand data: Panel E: Weak exogeneity test statistics
Variable  (m  p)  Δp  R  Δp^{max}  Δe  Δ^{2}p^{pos} 

χ^{2}(1)  6.57*  3.70^{+}  9.89**  12.6**  4.58*  0.59 
Table 2: A cointegration analysis of the Argentine money demand data: Panel F: Statistics for testing the significance of a given variable in β'x
Variable  (m  p)  Δp  R  Δp^{max}  Δe  Δ^{2}p^{pos}  trend 

χ^{2}(1)  2.71^{+}  2.79^{+}  7.65**  3.89*  7.44**  16.7**  1.31 
Table 2: A cointegration analysis of the Argentine money demand data: Panel G: Multivariate statistics for testing trend stationarity
Variable  (m  p)  Δp  R  Δp^{max}  Δe  Δ^{2}p^{pos} 

χ^{2}(5)  48.4**  45.7**  43.7**  56.6**  30.6**  24.2** 
Note: A box surrounds the first row of numbers in Panel C, and another box surrounds the first column of numbers in Panel D.
Table 2 also reports the standardized eigenvectors and adjustment coefficients, denoted and in a common notation. The first row of is the estimated cointegrating vector, which can be written in the form of (2):
(6) 
All coefficients have their anticipated signs. Also, the trend appears to be statistically insignificant: , where the asymptotic value is in square brackets. And, the hypothesis of "relative rates of return" in (3) appears acceptable. Numerically, the sum of the coefficients on and (17.06) is approximately equal to minus the coefficient on (17.53). Statistically, that restriction cannot be rejected: [0.850]. Jointly, the restrictions on the trend and rates of return also appear acceptable: [0.498].
Table 3 reports the estimated values of and when estimated unrestrictedly, and when estimated with a zero coefficient on the trend imposed, with the hypothesis of "relative rates of return" imposed, and with both of those restrictions imposed. The similarity of coefficient estimates across the various restrictions points to the robustness of the results and is partial evidence in favor of those restrictions.
Table 3: Justidentified and overidentified estimates of β and α, with corresponding estimated standard errors, from a cointegration analysis of Argentine money demand.: Panel A: Variable corresponding to an element of β'
Estimate of β'  m  p  Δp  R  Δp^{max}  Δe  Δ^{2}p^{pos}  trend 

Justidentified  1  10.89 (3.72)  17.53 (4.48)  1.20 (0.32)  6.17 (1.55)  62.69 (10.68)  0.0028 (0.0020) 
Zero coefficient on trend imposed  1  10.45 (2.71)  14.49 (3.27)  0.92 (0.14)  3.51 (1.07)  45.28 (7.75)  0 
Ratesofreturn restriction imposed  1  10.64 (3.55)  16.60 (3.87)  1.21 (0.27)  5.96 (1.46)  58.67 (8.14)  0.0027 (0.0019) 
Trend and ratesofreturn restrictions imposed  1  10.19 (2.56)  13.58 (2.76)  0.94 (0.12)  3.38 (1.01)  41.48 (5.51)  0 
Table 3: Justidentified and overidentified estimates of β and α, with corresponding estimated standard errors, from a cointegration analysis of Argentine money demand.: Panel B: Variable corresponding to an element of α'
Estimate of α'  m  p  Δp  R  Δp^{max}  Δe  Δ^{2}p^{pos} 

Justidentified  0.020 (0.007)  0.015 (0.007)  0.034 (0.012)  0.016 (0.004)  0.048 (0.021)  0.005 (0.005) 
Zero coefficient on trend imposed  0.024 (0.010)  0.016 (0.009)  0.050 (0.016)  0.020 (0.005)  0.051 (0.029)  0.003 (0.007) 
Ratesofreturn restriction imposed  0.021 (0.008)  0.015 (0.007)  0.036 (0.012)  0.017 (0.004)  0.050 (0.022)  0.005 (0.005) 
Trend and ratesofreturn restrictions imposed  0.026 (0.011)  0.017 (0.010)  0.054 (0.017)  0.022 (0.006)  0.054 (0.031)  0.004 (0.007) 
Thus, the nominal interest rate and inflation enter the longrun money demand function as the ex post real rate, with a semielasticity of about eleven, which is about unity at annual rates. The nominal interest rate relative to the exchangerate depreciation has about half that effect on money demand. Money demand is highly sensitive to the movement of inflation, both through and through the ratchet variable . In particular, for each additional percent in the maximal monthly inflation rate over the relative past, the coefficient on implies approximately one percent lower money holdings.
Figure 3 plots key aspects of equation (6)namely, the relationship between the variables , , and . Real money holdings fall as increases and as the return on money relative to goods declines.
Figure 3: The logarithm of real money (m  p), plotted against the maximal inflation rate (Δp^{max}) and the real interest rate (R  Δp)
Data for Figure 3
Date  (m  p)  Δp^{max}  R  Δp 

19771  2.404471     
19772  2.425484  0.077387  0.003187 
19773  2.447058  0.077387  0.004326 
19774  2.539662  0.077387  0.002530 
19775  2.624674  0.077387  0.001284 
19776  2.641501  0.077387  0.009510 
19777  2.660990  0.077387  0.004118 
19778  2.635934  0.107631  0.034631 
19779  2.623017  0.107631  0.001328 
197710  2.571095  0.118482  0.025682 
197711  2.559429  0.118482  0.013925 
197712  2.646366  0.118482  0.034037 
19781  2.608390  0.125444  0.026144 
19782  2.643875  0.125444  0.019156 
19783  2.621147  0.125444  0.020800 
19784  2.612881  0.125444  0.037837 
19785  2.622747  0.125444  0.014095 
19786  2.681274  0.125444  0.009325 
19787  2.699209  0.125444  0.004702 
19788  2.711565  0.125444  0.007451 
19789  2.711735  0.125444  0.000529 
197810  2.684952  0.125444  0.027872 
197811  2.680808  0.125444  0.017172 
197812  2.693516  0.125444  0.016770 
19791  2.662952  0.125444  0.052828 
19792  2.675306  0.125444  0.007481 
19793  2.680873  0.125444  0.011602 
19794  2.696720  0.125444  0.003212 
19795  2.721186  0.125444  0.001712 
19796  2.730533  0.125444  0.025969 
19797  2.744375  0.125444  0.000483 
19798  2.725727  0.125444  0.035226 
19799  2.743850  0.125444  0.007371 
197910  2.811398  0.125444  0.029475 
197911  2.840601  0.125444  0.011902 
197912  2.890433  0.125444  0.014715 
19801  2.889875  0.125444  0.011835 
19802  2.899849  0.125444  0.000573 
19803  2.888501  0.125444  0.007779 
19804  2.853743  0.125444  0.015198 
19805  2.817130  0.125444  0.010837 
19806  2.825433  0.125444  0.002445 
19807  2.857974  0.125444  0.015477 
19808  2.886855  0.125444  0.016281 
19809  2.884430  0.125444  0.001110 
198010  2.850921  0.125444  0.030220 
198011  2.846330  0.125444  0.000486 
198012  2.886839  0.125444  0.016854 
19811  2.850673  0.125444  0.008521 
19812  2.820511  0.125444  0.025366 
19813  2.770487  0.125444  0.023033 
19814  2.760892  0.125444  0.001149 
19815  2.719500  0.125444  0.007565 
19816  2.703099  0.125444  0.011884 
19817  2.703186  0.125444  0.010680 
19818  2.705274  0.125444  0.026472 
19819  2.735503  0.125444  0.014655 
198110  2.749043  0.125444  0.012913 
198111  2.738920  0.125444  0.004313 
198112  2.768634  0.125444  0.014965 
19821  2.714624  0.125444  0.040043 
19822  2.730076  0.125444  0.019802 
19823  2.738028  0.125444  0.022413 
19824  2.762294  0.125444  0.040980 
19825  2.791927  0.125444  0.043854 
19826  2.763377  0.125444  0.017394 
19827  2.657165  0.150646  0.099546 
19828  2.526791  0.150646  0.087088 
19829  2.395635  0.157750  0.087950 
198210  2.354903  0.157750  0.049566 
198211  2.334181  0.157750  0.022664 
198212  2.405781  0.157750  0.016032 
19831  2.420155  0.157750  0.043417 
19832  2.375893  0.157750  0.022496 
19833  2.355309  0.157750  0.006762 
19834  2.364466  0.157750  0.002247 
19835  2.398701  0.157750  0.013223 
19836  2.371626  0.157750  0.057891 
19837  2.354840  0.157750  0.012373 
19838  2.313019  0.159084  0.043084 
19839  2.243063  0.193652  0.051652 
198310  2.244771  0.193652  0.011777 
198311  2.253208  0.193652  0.030918 
198312  2.353592  0.193652  0.017977 
19841  2.414872  0.193652  0.002886 
19842  2.409003  0.193652  0.056938 
19843  2.359442  0.193652  0.084467 
19844  2.314524  0.193652  0.039721 
19845  2.285799  0.193652  0.027681 
19846  2.294710  0.193652  0.034766 
19847  2.292801  0.193652  0.012906 
19848  2.255929  0.205770  0.050770 
19849  2.144309  0.243317  0.088317 
198410  2.086846  0.243317  0.006660 
198411  2.141538  0.243317  0.030461 
198412  2.151631  0.243317  0.009637 
19851  2.109829  0.243317  0.049176 
19852  2.093735  0.243317  0.007874 
19853  2.012384  0.243317  0.035065 
19854  1.991975  0.258244  0.018244 
19855  2.015876  0.258244  0.075927 
19856  2.016300  0.266472  0.106472 
19857  2.092957  0.266472  0.025098 
19858  2.131651  0.266472  0.004822 
19859  2.185248  0.266472  0.015243 
198510  2.247214  0.266472  0.011723 
198511  2.256860  0.266472  0.007591 
198512  2.346090  0.266472  0.000213 
19861  2.363749  0.266472  0.001154 
19862  2.404743  0.266472  0.014243 
19863  2.393673  0.266472  0.014415 
19864  2.399513  0.266472  0.015258 
19865  2.421674  0.266472  0.008479 
19866  2.443864  0.266472  0.011451 
19867  2.441587  0.266472  0.030435 
19868  2.389271  0.266472  0.033201 
19869  2.366525  0.266472  0.024815 
198610  2.421847  0.266472  0.008759 
198611  2.432729  0.266472  0.003403 
198612  2.521382  0.266472  0.008688 
19871  2.505292  0.266472  0.017874 
19872  2.474059  0.266472  0.003000 
19873  2.467442  0.266472  0.048805 
19874  2.483266  0.266472  0.008908 
19875  2.505608  0.266472  0.006099 
19876  2.509485  0.266472  0.011974 
19877  2.494367  0.266472  0.021398 
19878  2.418701  0.266472  0.033539 
19879  2.397749  0.266472  0.000524 
198710  2.337599  0.266472  0.043538 
198711  2.348982  0.266472  0.008750 
198712  2.449926  0.266472  0.090546 
19881  2.436886  0.266472  0.044973 
19882  2.425096  0.266472  0.033768 
19883  2.435608  0.266472  0.018510 
19884  2.386193  0.266472  0.003016 
19885  2.376793  0.266472  0.026985 
19886  2.394030  0.266472  0.029785 
19887  2.346610  0.266472  0.001259 
19888  2.318104  0.266472  0.135927 
19889  2.361228  0.266472  0.019580 
198810  2.367192  0.266472  0.006885 
198811  2.437168  0.266472  0.046430 
198812  2.565286  0.266472  0.055841 
19891  2.648490  0.266472  0.035539 
19892  2.643205  0.266472  0.098414 
19893  2.608526  0.266472  0.058953 
19894  2.502546  0.287976  0.159024 
19895  2.281421  0.579239  0.574761 
19896  2.199170  0.763017  0.605983 
19897  1.827017  1.087311  0.748311 
19898  2.055586  1.087311  0.193083 
19899  2.193367  1.087311  0.015435 
198910  2.307980  1.087311  0.006558 
198911  2.321008  1.087311  0.032836 
198912  2.034576  1.087311  0.216007 
19901  1.318222  1.087311  0.319354 
19902  1.123179  1.087311  0.118771 
19903  0.742464  1.087311  0.214521 
19904  1.064193  1.087311  0.008289 
19905  1.200506  1.087311  0.039579 
19906  1.271376  1.087311  0.009868 
19907  1.350697  1.087311  0.008214 
19908  1.298890  1.087311  0.044694 
19909  1.236619  1.087311  0.021365 
199010  1.284640  1.087311  0.034909 
199011  1.342745  1.087311  0.007029 
199012  1.448775  1.087311  0.021298 
19911  1.453322  1.087311  0.061733 
19912  1.274226  1.087311  0.071041 
19913  1.269363  1.087311  0.011159 
19914  1.336993  1.087311  0.039441 
19915  1.385660  1.087311  0.012159 
19916  1.413816  1.087311  0.013760 
19917  1.427194  1.087311  0.007580 
19918  1.460799  1.087311  0.001075 
19919  1.486950  1.087311  0.006840 
199110  1.524018  1.087311  0.002903 
199111  1.576481  1.087311  0.007008 
199112  1.659075  1.087311  0.007018 
19921  1.680185  1.087311  0.018537 
19922  1.685963  1.087311  0.011312 
19923  1.681282  1.087311  0.011779 
19924  1.714833  1.087311  0.002785 
19925  1.786803  1.087311  0.002292 
19926  1.849497  1.087311  0.000190 
19927  1.880951  1.087311  0.007140 
19928  1.896892  1.087311  0.005855 
19929  1.912783  1.087311  0.001285 
199210  1.911993  1.087311  0.003580 
199211  1.919925  1.087311  0.005401 
199212  1.946672  1.087311  0.008167 
19931  1.987473  1.087311  0.000711 
19932    1.087311  0.003222 
19933    1.087311   
19934    1.087311   
19935    1.087311   
19936    1.087311   
19937    1.087311   
19938    1.087311   
19939    1.087311   
199310    1.087311   
199311    1.087311   
199312    1.087311   
Returning to Table 2, the coefficients in the first column of measure the feedback effects of the (lagged) disequilibrium in the cointegrating relation on the variables in the vector autoregression. Specifically, 0.020 is the estimated feedback coefficient for the money equation. The negative coefficient implies that lagged excess money induces smaller holdings of current money. The coefficient's numerical value implies slow adjustment to remaining disequilibrium. The estimated coefficient is numerically smaller than those for quarterly broad money demand (e.g., 0.26, 0.15, and 0.20 in Taylor (1986)) and monthly currency demand (e.g., 0.14 for Argentina in Ahumada (1992)). However, smaller adjustment coefficients are plausible with highfrequency data for a broad aggregate.
The third block from the bottom of Table 2 reports values of the statistic for testing weak exogeneity of a given variable for the cointegrating vector. Equivalently, the statistic tests whether or not the corresponding row of is zero; see Johansen (1992a, 1992b). If the row of is zero, disequilibrium in the cointegrating relationship does not feed back onto that variable. Surprisingly, inflation (including in its form ) may be weakly exogenous. However, the interest rate, the exchange rate, and the ratchet variable are not weakly exogenous, justifying a systems approach to analyzing cointegration.
The penultimate block in Table 2 reports statistics for testing the significance of individual variables in the cointegrating vector. Each variable is significant, except the linear trend.
The final block in Table 2 reports values of a multivariate statistic for testing the trend stationarity of a given variable. Specifically, this statistic tests the restriction that the cointegrating vector contains all zeros except for a unity corresponding to the designated variable and an unrestricted coefficient on the trend, with the test being conditional on the presence of exactly one cointegrating vector; see Johansen (1995), p. 74]. For instance, the null hypothesis of trendstationary real money implies that the cointegrating vector is (1 0 0 0 0 0 *)′, where "*" represents an unrestricted coefficient on the linear trend. Empirically, all of the stationarity tests reject with pvalues less than 0.1%. By being multivariate, these statistics may have higher power than their univariate counterparts. Also, the null hypothesis is the stationarity of a given variable rather than the nonstationarity thereof, and stationarity may be a more appealing null hypothesis. That said, these rejections of stationarity are in line with the inability in Table 1 to reject the null hypothesis of a unit root in each of , , , , and .
Because , , and are not weakly exogenous for the cointegrating vector, inferences in a single equation for broad money could be hazardous if the cointegrating vector is estimated jointly with the equation's dynamics; see Hendry (1995). One solution is to model a subsystem. A second solution is to construct an error correction term from the system estimates and then develop a single equation ECM that uses that systembased error correction term. A third solutionadopted belowis to develop a single equation ECM from the single equation ADL, noting that the system estimate of the cointegrating relationship is numerically close to the ADL's longrun solution. See Hendry and Doornik (1994) and Juselius (1992) for paradigms of the first two approaches.
This section first describes the model selection algorithms in PcGets and Autometrics (Section 5.1) and then applies these algorithms to an ECM representation of an ADL for Argentine money demand (Section 5.2).
Hendry and Krolzig (2001) develop a computer program PcGets, which extends and improves upon Hoover and Perez's (1999) automated modelselection algorithm; see also Hendry and Krolzig (1999, 2003, 2005) and Krolzig and Hendry (2001). Doornik and Hendry (2007) implement a thirdgeneration algorithm called Autometrics, which is part of PcGive version 12. PcGets and Autometrics utilize onestep and multistep simplifications along multiple paths, diagnostic tests as additional checks on the simplified models, and encompassing tests to resolve multiple terminal models. Both analytical and Monte Carlo evidence show that the resulting model selection is relatively nondistortionary for Type I errors. At an intuitive level, PcGets and Autometrics function as a series of sieves that aim to retain parsimonious congruent models while discarding both noncongruent models and overparameterized congruent models. This feature of the algorithms is eminently sensible, noting that the data generation process itself is congruent and is as parsimonious as feasible.
The remainder of the current subsection summarizes PcGets and Autometrics as automated modelselection algorithms, thereby providing the necessary background for interpreting their application in Section 5.2. For ease of reference, the algorithm in PcGets is divided into four "stages", denoted Stage 0, Stage 1, Stage 2, and Stage 3. For full details of PcGets's algorithm, see Hendry and Krolzig (2001), Appendix A1). Hendry and Krolzig (2003) describe the relationship of the generaltospecific approach to other modeling approaches in the literature, and Hoover and Perez (2004) extend the generaltospecific approach to crosssection regressions.
Stage 0: the general model and F presearch tests. Stage 0 involves two parts: the estimation and evaluation of the general model, and some presearch tests aimed at simplifying the general model before instigating formal multipath searches.
First, the general model is estimated, and diagnostic statistics are calculated for it. If any of those diagnostic statistics is unsatisfactory, the modeler must decide what to do nextwhether to "go back to the drawing board" and develop another general model, or whether to continue with the simplification procedure, perhaps ignoring the offending diagnostic statistic or statistics.
Second, PcGets attempts to drop various sets of potentially insignificant variables. PcGets does so by dropping all variables at a given lag, starting with the longest lag. PcGets also does so by ordering the variables by the magnitude of their tratios and either dropping a group of individually insignificant variables or (alternatively) retaining a group of individually statistically significant variables. In effect, an F presearch test for a group of variables is a single test for multiple simplification paths, a characteristic that helps control the costs of search. If these tests result in a statistically satisfactory reduction of the general model, then that new model is the starting point for Stage 1. Otherwise, the general model itself is the starting point for Stage 1.
Stage 1: a multipath encompassing search. Stage 1 tries to simplify the model from Stage 0 by searching along multiple paths, all the while ensuring that the diagnostic tests are not rejected. If all variables are individually statistically significant, then the initial model in Stage 1 is the final model. If some variables are statistically insignificant, then PcGets tries deleting those variables to obtain a simpler model. PcGets proceeds down a given simplification path only if the models along that path have satisfactory diagnostic statistics. If a simplification is rejected or if a diagnostic statistic fails, PcGets backtracks along that simplification path to the most recent previous acceptable model and then tries a different simplification path. A terminal model results if the model's diagnostic statistics are satisfactory and if no remaining regressors can be deleted.
If PcGets obtains only one terminal model, then that model is the final model, and PcGets proceeds to Stage 3. However, because PcGets pursues multiple simplification paths in Stage 1, PcGets may obtain multiple terminal models. To resolve such a situation, PcGets creates a union model from those terminal models and tests each terminal model against that union model. PcGets then creates a new union model, which nests all of the surviving terminal models; and that union model is passed on to Stage 2.
Stage 2: another multipath encompassing search. Stage 2 in effect repeats Stage 1, applying the simplification procedures from Stage 1 to the union model obtained at the end of Stage 1. The resulting model is the final model. If Stage 2 obtains more than one terminal model after applying encompassing tests, then the final model is selected by using the Akaike, Schwarz, and HannanQuinn information criteria. See Akaike (1973, 1981), Schwarz (1978), and Hannan and Quinn (1979) for the design of these information criteria, and Atkinson (1981) for the relationships between them.
Stage 3: subsample evaluation. Stage 3 reestimates the final model over two subsamples and reports the results. If a variable is statistically significant in the full sample and in both subsamples, then the inclusion of that variable in the final model is regarded as "100% reliable". If a variable is statistically insignificant in one or both subsamples or in the full sample, then its measure of reliability is reduced. A variable that is statistically insignificant in both subsamples and in the full sample is regarded as being "0% reliable". The modeler is left to decide what action, if any, to take in light of the degree of reliability assigned to each of the regressors.
(a) presearch simplification of the general unrestricted model (Stage 0),
(b) multipath (and possibly iterative) selection of the final model (Stages 1 and 2), and
(c) postsearch evaluation of the final model (Stage 3).
This subsection's description of these four stages summarizes the algorithm in PcGets. Below, Section 5.2 summarizes the actual simplifications found by PcGets in practice, thereby providing additional insight into PcGets's algorithm.
PcGets requires the modeler to choose which tests are calculated and to specify the critical values for those tests. In PcGets, the modeler can choose the test statistics and their critical values directly, although doing so is tedious because of the number of statistics involved. To simplify matters, PcGets offers two options with predesignated selections of test statistics and critical values. These two options are called "liberal" and "conservative" model selection strategies. The liberal strategy errs on the side of keeping some variables, even although they may not actually matter. The conservative strategy keeps only variables that are clearly significant statistically, erring in the direction of excluding some variables, even although those variables may matter. Which strategy is preferable depends in part on the data themselves and in part on the objectives of the modeling exercise, although (as below) the two approaches may generate similar or identical results.
The algorithm in PcGets is generaltospecifc, multipath, iterative, and encompassing, with diagnostic tests providing additional assessments of statistical adequacy, and with options for presearch simplification. The algorithm in Autometrics shares these characteristics with the algorithm in PcGets; hence, many of the remarks above about PcGets apply directly to Autometrics. However, Autometrics (unlike PcGets) uses a tree search method, with refinements on presearch simplification and on the objective function. See Doornik and Hendry (2007) and Doornik (2008) for details.
Using PcGets and Autometrics, the current subsection assesses the possible path dependence of equation (4). The initial general model is estimated; and the algorithms simplify that general model under each of the permutations implied by the list of choices below. While the algorithms do obtain multiple distinct final models, equation (4)or simple variants of itappears statistically sensible; and one variant obtained by Autometrics is even more parsimonious than (4). These results bolster the model design in Kamin and Ericsson (1993) and offer an improvement on it.
The multipath searches in PcGets and Autometrics allow
investigation of equation (4)'s
robustness and examination of the empirical properties of the two
algorithms themselves. In addition, four choices within the model
selection process permit further insights. In PcGive, these choices
concern the following.
For model strategy (choice #1), the options in Autometrics do not correspond precisely to PcGets's liberal and conservative strategies. Instead, Autometrics allows the user to select a "target size", which is meant to equal "the proportion of irrelevant variables that survives the [simplification] process" (Doornik, 2008). In the analysis below, Autometrics's target size is either 5% or 1%, which appear to approximate liberal and conservative strategies in PcGets. For presearch testing (choice #2), the selected option in Autometrics is either presearch for both variable reduction and lag reduction, or no presearch for eitherin order to match PcGets as closely as possible. The third choice above is identical for PcGets and Autometrics, as is the fourth choice.
For both PcGets and Autometrics, the third choice (the representation of the initial general ECM) can affect the final model selected. In simplifying the initial model, PcGets and Autometrics impose only "zero restrictions", i.e., the algorithms can set coefficients to be equal only to zero. Although a linear model is invariant to nonsingular linear transformations of its data, the coefficients of that model are not invariant to such transformations. For example, a model with regressors and is invariant to including the regressors and instead; but the deletion of results in two different simplifications, depending on the representation. See Campos and Ericsson (1999) for additional discussion.
Table 4 lists the estimates and standard errors for the ECM representation of the unrestricted seventhorder ADL model of , , , , , and . The standard diagnostic statistics do not reject. The implied coefficient on the error correction term appears to be highly significant statistically, with a tratio of 3.53. The intermediate ADL (in ECM representation) is Table 4, but reestimated with the "boxedin" coefficients in Table 4 set to zero. Kamin and Ericsson (1993) show that the estimated coefficients in this intermediate model are close to those in the unrestricted ECM in Table 4; and the intermediate ADL is a statistically acceptable reduction of Table 4, with . For ease of reference, the intermediate ADL is denoted Table 4*.
Table 4: An unrestricted error correction representation for real money conditional on inflation, the interest rate, and the change in the exchange rate: Panel A: Regression coefficients and estimated standard errors.
Variable^{a,b,c}  Lag j = 0 
Lag j = 1 
Lag j = 2 
Lag j = 3 
Lag j = 4 
Lag j = 5 
Lag j = 6 
Lag j = 7 

Δ(m  p)_{tj}  1 () 
0.270 (0.088) 
0.154 (0.093) 
0.019 (0.094) 
0.029 (0.079) 
0.204 (0.079) 
0.125 (0.072) 
 
Δp_{tj}  0.768 (0.101) 
0.177 (0.149) 
0.041 (0.140) 
0.144 (0.136) 
0.121 (0.117) 
0.168 (0.119) 
0.275 (0.122) 
0.053 (0.079) 
ΔR_{tj}  0.222 (0.053) 
0.031 (0.159) 
0.172 (0.149) 
0.140 (0.143) 
0.039 (0.128) 
0.002 (0.096) 
0.131 (0.090) 
 
0.223 (0.155) 
0.169 (0.184) 
0.327 (0.281) 
0.599 (0.297) 
0.104 (0.268) 
0.211 (0.264) 
0.132 (0.317) 
 

Δ  0.406 (0.162) 
0.049 (0.154) 
0.180 (0.161) 
0.114 (0.156) 
0.168 (0.133) 
0.003 (0.137) 
0.181 (0.130) 
0.017 (0.112) 
Δe_{tj}  0.004 (0.020) 
0.040 (0.022) 
0.014 (0.023) 
0.025 (0.022) 
0.040 (0.022) 
0.011 (0.023) 
0.032 (0.023) 
0.032 (0.022) 
(m  p)_{tj}   
0.053 (0.015) 
 
 
 
 
 
 
R_{tj}   
0.441 (0.158) 
 
 
 
 
 
 
 
0.055 (0.019) 
 
 
 
 
 
 

B_{tj}  0.353 (0.139) 
0.178 (0.078) 
0.058 (0.073) 
0.286 (0.084) 
 
 
 
 
S_{tj}  0.160 (0.047) 
 
1.62 (1.20) 
0.08 (1.21) 
0.41 (1.10) 
0.21 (1.16) 
2.81 (1.02) 
 
S_{tj6}   
0.06 (0.97) 
0.10 (1.03) 
0.55 (1.01) 
0.86 (1.03) 
0.20 (1.02) 
4.35 (1.02) 
 
Table 4: An unrestricted error correction representation for real money conditional on inflation, the interest rate, and the change in the exchange rate: Panel B: Regression statistics.
Notes:
a. The dependent variable is Δ(m  p)_{t}. Even so, the equation is in
levels, not in differences, noting the inclusion of the
regressor (m  p)_{t1}.
b. The variables {S_{ti}} are the seasonal
dummies, except that S_{0} is the intercept.
February is S_{t2}, March is S_{t3}, etc. For readability, the coefficients and estimated
standard errors for the seasonal dummies have been multiplied by
100.
c. The 33 coefficients that are "boxed in" are set equal to zero
in the partially restricted intermediate error correction
representation denoted Table 4*.
d. The statistic LM_{p} is the Lagrange multiplier
statistic for testing the imposed restriction of longrun price
homogeneity.
e. A box surrounds the coefficients corresponding to variables dropped in Table 4*. Those variables are Δ(m  p)_{tj} for j=2,3,4; Δp_{tj} for j=2,3,4,7; ΔR_{tj} for j=2,3,4,5,6; Δ(Δp_{tj}^{max}) for j=0,1,2,3,4,5,6; Δ^{2}p_{tj}^{pos} for j=1,2,3,4,5,6,7; and Δe_{tj} for j = 0,2,3,4,5,6,7.
Table 5 summarizes PcGets's model simplifications under the 24 different scenarios described above; Table 6 does likewise for Autometrics. In these tables, is the number of regressors in the general model for multipath searches, is the number of coefficients in the final specific model for multipath searches, the "number of paths" is the number of different simplification paths considered in a multipath search, the "number of terminal models" is the number of distinct terminal specifications after a multipath search, and is the residual standard error of the final specific model. If multipath searches are iterated, the table lists values for each iteration, where appropriate. The "number of models estimated" is the total number of distinct models estimated in the multipath search.
Table 5: Statistics on computerautomated model selection by PcGets of models for Argentine money demand, categorized according to model strategy, presearch testing, representation of the general model, and choice of general model: Panel A: The general model is Table 4 or equivalent
Model Strategy  Presearch?  Representation?  k_{1}  k_{f}  Number of paths  Number of terminal models  (%) 

L  No  Table 4  63, 31  19  55, 7  13, 1  2.132 
L  No  Nested  63, 31, 26  24  56, 18, 9  9, 3, 5  1.952 
L  No  Fixed  63, 25  22  53, 10  6, 3  1.954 
L  Yes  Table 4  26  21  10  3  2.078 
L  Yes  Nested  28  23  11  3  1.989 
L  Yes  Fixed  23  22  6  2  1.986 
C  No  Table 4  63, 32, 31  23  61, 21, 20  10, 5, 4  2.015 
C  No  Nested  63, 31  21  61, 22  9, 5  2.007 
C  No  Fixed  63, 24  21  55, 9  6, 3  1.988 
C  Yes  Table 4  21  21  1  1  2.139 
C  Yes  Nested  22, 21  21  10, 8  2, 1  2.073 
C  Yes  Fixed  24, 18  18  11, 3  1, 1  2.137 
Table 5: Statistics on computerautomated model selection by PcGets of models for Argentine money demand, categorized according to model strategy, presearch testing, representation of the general model, and choice of general model: Panel B: The general model is Table 4* or equivalent
Model Strategy  Presearch?  Representation?  k_{1}  k_{f}  Number of paths  Number of terminal models  (%) 

L  No  Table 4*  30  20  18  1  2.149 
L  No  Nested  30  18  20  1  2.137 
L  No  Fixed  30  18  20  1  2.137 
L  Yes  Table 4*  20  20  1  1  2.149 
L  Yes  Nested  18  18  1  1  2.137 
L  Yes  Fixed  18  18  1  1  2.137 
C  No  Table 4*  30, 20  20  19, 3  1, 1  2.149 
C  No  Nested  30, 18  18  22, 3  2, 1  2.137 
C  No  Fixed  30, 18  18  22, 3  2, 1  2.137 
C  Yes  4*  20, 20  20  3, 3  1, 1  2.149 
C  Yes  Nested  18, 18  18  3, 3  1, 1  2.137 
C  Yes  Fixed  18, 18  18  3, 3  1, 1  2.137 
Table 6: Statistics on computerautomated model selection by Autometrics of models for Argentine money demand, categorized according to target size, presearch testing, representation of the general model, and choice of general model: Panel A: The general model is Table 4 or equivalent
Target Size  Presearch?  Representation?  k_{1}  k_{f}  Number of Models Estimated  Number of terminal models  (%) 

5%  No  Table 4  63, 41  23  706  10, 17  1.997 
5%  No  Nested  63, 36  21  378  8, 13  1.978 
5%  No  Fixed  63, 31  20  306  6, 7  2.003 
5%  Yes  Table 4  57, 37  22  470  6, 12  2.008 
5%  Yes  Nested  50, 32  22  371  6, 7  1.972 
5%  Yes  Fixed  45, 30  22  255  8, 8  1.986 
1%  No  Table 4  63, 37  19  751  10, 20  2.095 
1%  No  Nested  63, 29  21  501  8, 10  1.978 
1%  No  Fixed  63, 22  18  497  2, 2  2.096 
1%  Yes  Table 4  41, 35  20  677  10, 20  2.072 
1%  Yes  Nested  39, 26  20  394  5, 5  2.014 
1%  Yes  Fixed  30, 23  19  168  4, 4  2.078 
Table 6: Statistics on computerautomated model selection by Autometrics of models for Argentine money demand, categorized according to target size, presearch testing, representation of the general model, and choice of general model: Panel B: The general model is Table 4* or equivalent
Target Size  Presearch?  Representation?  k_{1}  k_{f}  Number of Models Estimated  Number of terminal models  (%) 

5%  No  Table 4*  30, 20  20  46  1, 1  2.149 
5%  No  Nested  30, 18  18  65  1, 1  2.137 
5%  No  Fixed  30, 18  18  65  1, 1  2.137 
5%  Yes  Table 4*  25, 20  20  36  1, 1  2.149 
5%  Yes  Nested  23, 18  18  40  1, 1  2.137 
5%  Yes  Fixed  23, 18  18  43  1, 1  2.137 
1%  No  Table 4*  30, 18  18  77  1, 1  2.211 
1%  No  Nested  30, 14  14  139  1, 1  2.236 
1%  No  Fixed  30, 16  16  2  1  2.192 
1%  Yes  Table 4*  21, 20  20  30  1, 1  2.149 
1%  Yes  Nested  19, 17  17  36  1, 1  2.168 
1%  Yes  Fixed  19, 17  17  38  1, 1  2.168 
Note: A box surrounds the eighth row of entries in the second panel.
Several features of the simplifications in Tables 5 and 6 are notable. First, presearch testing typically reduces the number of paths that need to be searched in Stage 1, and often markedly so. As a consequence, presearch testing frequently reduces the number of multiple terminal models and, in some instances, obtains the final model. Second, if the initial general model is the intermediate ECM (Table 4*, rather than the general ECM in Table 4), that choice is in effect a presearch, albeit an informal one. That choice also typically obtains a single terminal model on the initial multipath search. Third, a conservative strategy generally obtains a more parsimonious model than a liberal strategy, as expected. Fourth, Kamin and Ericsson's (1993) model results from a conservativelike strategy, as is apparent from examining the specifications of the final models in Tables 5 and 6. Fifth, the 1% and 5% target sizes in Autometrics appear closely comparable to the liberal and conservative strategies in PcGets. That said, in several instances, Autometrics dominates PcGets by obtaining a more parsimonious model with a better fit (in terms of ), whereas PcGets never dominates Autometrics in that sense. This outcome reflects differences in the algorithms' details. Finally, data transformations through the "nesting" approach permit a final representation that is more highly parsimonious than previously obtained; see the boxedin result for Autometrics on Table 6.
The corresponding model, which improves on equation (4), is as follows.
Δ  (7)  
The coefficients in equation (7) are little changed from the corresponding ones in equation (4), except that the coefficients for and are restricted to be zero. No tests reject at the 1% level (an implication of choices made in the algorithm's parameters), although some do at the 5% level. Equation (7) has virtually the same economic interpretation as equation (4), and it is more parsimonious than (4). PcGets and Autometrics thus verify the robustness of equation (4)'s specification, and Autometrics improves upon that specification.
Computerautomated model selection with the software packages PcGets and Autometrics demonstrates the robustness of Kamin and Ericsson's (1993) final error correction model and improves on it by using multipath searches that would be tedious and prohibitively timeconsuming with standard econometrics packages. Longrun money demand is driven by a negative ratchet effect from inflation, and by the opportunity cost of holding pesodenominated financial assets rather than Argentine goods or U.S. dollars. Shortrun dynamics are consistent with an Sstype inventory model that is interpretable as having either real or nominal shortrun bounds.
Several general remarks are germane, and each suggests extensions to the current analysis. First, improvements to the model selection algorithms may and do obtain an improved model specification. Computerautomated modelselection algorithms are still in their youthif not in their infancyand considerable analytical, Monte Carlo, and empirical research is ongoing; see Hendry and Krolzig (1999, 2003, 2005), Krolzig and Hendry (2001), Hoover and Perez (2004), Doornik (2008), Hendry, Johansen, and Santos (2008), Hoover, Demiralp, and Perez (2008), Hoover, Johansen, and Juselius (2008), and Johansen and Nielsen (2008).
Second, insights by other researchers may improve the current model in a progressive research strategy. For example, Nielsen (2004), building on Hendry, von UngernSternberg (1981), proposes an alternative measure of the opportunity cost of holding money that may better capture agents' behavior in a hyperinflationary environment. Preliminary tests for that alternative measure as an omitted variable in Table 4 do not reveal an improved specification, however. For instance, for a variable in levels, define as , which is 's percentage change, measured as a fraction. Omitted variables tests include for , for , and for . None of these tests reject at standard levels. Still, Table 4 is a relatively unrestricted model, so further investigation is merited, particularly because differs substantially from at high inflation rates and hence the interpretation of may be affected.
Third, Kongsted (2005) develops a procedure for testing the nominaltoreal transformation, which is only informally investigated herein for money by using the ADF statistics. Fourth, in the VAR, the variables and are transformations of , so further consideration of their joint distributional properties is desirable. Fifth, data observations after 1993 may be informative. Even so, mechanistic extensions of the existing data may not be sufficient, as when data definitions change, the array of available assets alters, and underlying economic conditions shift; see Ericsson, Hendry, and Prestwich (1998).
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* Forthcoming in Jennifer L. Castle and Neil Shephard (eds.) The Methodology and Practice of Econometrics: A Festschrift in Honour of David F. Hendry, Oxford University Press, Oxford, 2008. The authors are staff economists in the Division of International Finance, Board of Governors of the Federal Reserve System, Washinton D.C. 20551 U.S.A.; and they may be reached on the Internet at [email protected] and [email protected] respectivityl. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. The authors are grateful to Julia Campos, Jennifer Castle, Dale Henderson, David Hendry, Katarina Juselius, Jaime Marquez, Bent Nielsen, Anders Rahbek, Neil Shephard, and two referees for helpful comments. All numerical results were obtained using PcGiv Version 12.00, Autometrics Version 1.5, and PcGets Version 1.02; see Doornik and Hendry (2007), Doornik (2008), and Hendry and Krolzig (2001). Return to text
1. A preliminary investigation found little role for Y in the cointegration analysis or in error correction modeling. This is consistent with Ahumada's (1992) evidence on currency demand, and may be due to the relatively stationary nature of real GDP in Argentina over the sample period. Return to text
2. For further analytical and empirical discussion of the Argentine economy, see Howard (1987), the World Bank (1990), Kamin (1991), Kiguel (1991), Manzetti (1991), Beckerman (1992), Kamin and Ericsson (1993), and Helkie and Howard (1994). See Dominguez and Tesar (2007) for a history of the post1990 period. Return to text
3. Hendry and Ericsson (1991, p. 853] and Baba, Hendry, and Starr (1992) find similar results for narrow money demand in the United Kingdom and the United States. Also, in keeping with this observation about Δ^{2}p_{t}, Kamin and Ericsson (1993) simplify the restricted intermediate ADL to obtain an alternative ECM where that ECM has Δm_{t} as the dependent variable. Return to text
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