Estimating short and long-run relationships: a guide for the applied economist

Abstract

Many applied economists face problems in selecting an appropriate technique to estimate short and long-run relationships with the time series methods. This article reviews three alternative approaches viz., general to specific, vector autoregressions and the vector error correction models. As in other methodological controversies, definite answers are difficult. It is suggested that if these techniques are seen as tools to summarize data, as in Smith (2000), often there maybe only minor differences in their estimates. Therefore a computationally attractive technique is likely to be popular.

Acknowledgments

I wish to thank the editor of this journal, Professor Mark Taylor, for his constructive suggestions and my graduate students in EC405, Econometric Models and Forecasting, for their feedback.

Notes

¹ This way of looking at this controversy, from a philosophical perspective, can be justified with the following observation by Granger (1997).

 ‘… the actual economy appears to be very complicated, partly because it is the aggregation of millions of nonidentical, nonindependent decision-making units, … . A further practical problem is that the observation period of the data does not necessarily match the decision making periods (temporal aggregation). It can be argued that even though the quantity of data produced by a macroeconomy is quite large, it is still quite insufficient to capture all of the complexities of the DGP. The modelling objective has thus to be limited to providing an adequate or satisfactory approximation to the true DGP. Hopefully, as modelling technology improves and data increases, better approximations will be achieved, but actual convergence to the truth is highly unlikely.’ (p. 169, my italics). In light of this observation it may be said that no matter how complicated is our technique, we may never know the truth. Therefore, a simpler method with good and acceptable degree of precision is likely to be widely used.

² At the outset a limitation of this article should be noted. The widely used current techniques viz., GETS, VAR and VECM are all based on autoregressive (AR) formulations. This may not satisfy some specialist time series econometricians who favour autoregressive integrated moving average (ARIMA) or ARMA formulations; Harvey (1997). For a good survey of the historical developments in various econometric techniques see Section V in Pesaran and Smith (1992). They also briefly comment on the problems in using ARIMA formulations.

³ The general presumption that unit root tests based on endogenous breaks, e.g., Zivot and Andrews (1992), are better than tests with a known prior date, e.g., Perron (1989), is controversial and depends on ones methodological views. Maddala and Kim (1998) noted that:

There is a lot of work on testing with unknown switch points. In practice, there is a lot of prior information and there is no reason why we should not use it. For instance, suppose there is a drastic policy change or some major event (for example, oil price shock) that occured at time t ₀. It does not make sense to ask the question of whether there was a structural change around that period. It is not very meaningful to search for a break over the entire sample period ignoring this prior information. Maddala and Kim (1998, p. 398), our italics.

These observations imply that perhaps testing for unit roots with a priori known dates, e.g., Perron (1989), is more meaningful than endogenous switching points. Needless to say this is a philosophical issue and there are likely to be differences.

⁴ We are using a version GETS that was developed after the unit roots revolution.

⁵ This is also known as the unrestricted autoregressive distributed lag model (ARDL). In the ARDL approach, the lag structure is determined by an optimal lag search procedure. This procedure can be also used to determine the lag length in the GUM. Equation 2 in the text is based on transforming the following equation in the levels of the variables. For simplicity, only one lag is used for C.

This can be written as:

where λ = (1 − b ₁) and γ = (g ₀ + g ₁)/(1 − b ₁). Note that if (A) is solved for equilibrium, the coefficient of Y is the same as this. To obtain the equilibrium equation from (A), set C_t  = C_{t −1} and Y_t  = Y_{t −1} and solve. A trend variable may be also added to the ARDL.

⁶ It is of interest to note that ECM is very much a LSE concept and its usefulness has been illustrated in several applications. Professor W. B. Phillips seems to have first developed this concept to determine the adjustment needed in the policy instruments to maintain a target variable close to its desired value. It was later used in other applications, including in GETS, by Sargan, Hendry, Mizon and also Engle and Granger (1987) in developing their concept of cointegration; for a survey of ECM and its different interpretations see Alagoskoufis and Smith (1991). It may also be said that some LSE economists who were influenced by the methodological views of Karl Popper, were aware of the methodological conflict between testing equilibrium theories with data from a disequilibrium world.

⁷ A judicious application of the variable deletion test in the standard software like Microfit also gives good parsimonious equations. When the sample size is small, relative to the number of explanatory variables, using these variable deletion tests is a more practical option because PcGets is good when the GUM is adequately elaborate.

⁸ If the ECM of (15) is denoted as ECMM, its lagged value will either have a wrong sign and/or insignificant in the equations in which the dependent variable is ΔlnY_t or ΔR_t . This is an alternative and intuitively plausible approach for identification. Similarly, if the lagged ECMM term is significant but its coefficient is positive, for example, in the equation for ΔlnY_t , it is a clear indication that disequilibrium in money holdings affects income and therefore income is not an exogenous variable. These are just intuitive rules of thumb and help to understand what is behind more formal tests. For testing with these rules of thumb, GETS specifications are useful.

⁹ Rao and Singh have used this summary of facts and found that the demand for money in Fiji is temporally stable. Therefore, they have raised doubts on targeting the rate of interest by the Reserve Bank of Fiji.

¹⁰ If the number of CVs equals the number of variables, then the variables are stable in their levels i.e., I(0) and the equations can be estimated with the standard classical methods. If there is no cointegration vector, it implies that the underlying economic theory is inadequate and perhaps some other relevant variables are missing from the model.

¹¹ EViews has a few seasonal adjustment options including X12.

¹² The coefficients in the CVs in these relationships and other summary statistics are very similar, although there were differences in the structures of the short-run dynamic adjustments.

¹³ At the University of the South Pacific, Singh (2005) has developed with GETS a small structural forecasting model for Fiji with annual data from 1970–2002. Its dynamic simulation properties are impressive.