2.1 Analysis of time series  (Page 6/6)

Estimation of arma models

The estimation of ARMA models are relatively easy in Stata . The basic command to estimate an ARMA model is: .arima depvar [varlist], ar( numlist ) ma( numlist ) . ARIMA means A uto R egressive I ntegrated M oving A verage. See Enders (2005: 67) for a discussion of what integrated means. We can ignore it given our limited purposes. The first thing to notice in the command that this command can apply to either to a single variable or to an equation. If [varlist] is omitted, Stata will produce an estimate of the ARMA model for that variable; if the list is included, it will estimate the model with the disturbances allowed to have the ARMA structure specified in the command. Figure 14 reports the estimation of an ARMA model for real investment levels. Notice that we write AR(1/2) so that Stata knows to include both the first and second autoregressive term. A command of AR(2) would include only the second autoregressive term. In Figure 15 we report the ARMA (2, 1) estimation of (1).

The interpretation of these results is not obvious. We check the sensitivity of these results by estimation some other models. The results of these estimations are reported in Table 2 and Table 3. Based purely on ML tests, it would appear that AR(1) model in Table 2 is as good as any of the models describing the ARMA structure of real investments. On the other hand, the results reported in Table 3 suggests that the ARMA(2, 1) appears to be the best model to assume for the disturbance term in the estimates of Equation (1).

Other time-series concepts

There are a large number of additional time-series methods and issues that are not discussed in this module. These topics include, among others, ARCH and GARCH estimators, unit roots, the Dickey-Fuller test, and vector autoregression (VAR) models. There is no way to do justice to these topics in notes as short as these are. Moreover, it is necessary to discuss difference equations (the discrete version of differential equations) if one wants to understand many of these topic at anything more than an intuitive level. Those interested in these topics should enroll in the forecasting course (Economics 422) or, if they cannot, plan to read several textbooks on whatever econometric tool they need to understand.

Exercise

a. Create a graph of (a) the data set morntemp, (2) the autocorrelations of morntemp, and (3) the partial autocorrelations of morntemp (you will have to set the matrix size to some number greater than 43 using the command .set matsize # ).

b. Estimate the following models:

1. ARMA(2,2) for morntemp.
2. ARMA(2,2) for morntemp as a function of the season dummy variables.
3. ARMA(2,2) for morntemp as a function of the monthly dummy variables.
4. ARMA(2,2) for morntemp as a function of the monthly dummy variables and the annual dummy variables.
5. ARMA(1,2) for morntemp as a function of the monthly dummy variables and the annual dummy variables.
6. ARMA(1,1) for morntemp as a function of the monthly dummy variables and the annual dummy variables.

c. Arrange the parameter estimates in a table and comment on them. Include the results of estimating (6) using OLS; what is the DW-statistic for this regression?

References

Cochran, D. and G. Orcutt (1949). Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms. Journal of the American Statistical Association 44 : 32-61.

Enders, Walter (1995). Applied Econometric Time Series (New York: John Wiley&Sons, Inc.).

Greene, William H. (1990). Econometric Analysis (New York: Macmillan Publishing Company).

StataCorp (2003). Stata Statistical Software: Release 8.0: Stata Time-Series Reference Manual (College Station, TX: Stat Corporation).