Open the usa.dta data set (1984q1-2009q4), create the dates and declare it as a time series. Save the data so you won t have to do this step again.

Transcription

1 ARIMA forecasts Open the usa.dta data set (1984q1-2009q4), create the s and declare it as a time series. Save the data so you won t have to do this step again. use usa, clear * * Create s and declare time-series * generate = q(1984q1) + _n-1 format %tq tsset Here, we plot real GDP, its difference, its natural log and the log difference. qui gen lg = ln(gdp) qui tsline gdp, name(g, replace) qui tsline D.gdp, name(dg, replace) qui tsline lg, name(lg, replace) qui tsline D.lg, name(dlg, replace) graph combine g Dg lg Dlg

2 real US gross domestic product real US gross domestic product, D ln(gdp) D.ln(gdg) Looks like there is a trend in the level (perhaps exponential). The difference (upper right) may show a slight upward trend until the bottom dropped out in late Still, I see no reason to use logs, so I won t. Others might disagree. dfgls gdp dfgls D.gdp, notrend. dfgls gdp DF-GLS for gdp Number of obs = 91 Maxlag = 12 chosen by Schwert criterion DF-GLS tau 1% Critical 5% Critical 10% Critical [lags] Test Statistic Value Value Value

3 Opt Lag (Ng-Perron seq t) = 11 with RMSE Min SC = at lag 1 with RMSE Min MAIC = at lag 1 with RMSE That s not too good. Clearly we are in the not reject region. The level is nonstationary. And the differences with notrend results:. dfgls D.gdp, notrend DF-GLS for D.gdp Number of obs = 90 Maxlag = 12 chosen by Schwert criterion DF-GLS mu 1% Critical 5% Critical 10% Critical [lags] Test Statistic Value Value Value Opt Lag (Ng-Perron seq t) = 10 with RMSE Min SC = at lag 1 with RMSE Min MAIC = at lag 1 with RMSE The statistic is significant at every lag. Go for the differences. Removing the trend has no substantive effect in this case. I think the DF-GLS test is the way to go as opposed to the usual DF or ADF test (more powerful than ADF) so I ll use it. Also, this test in Stata is useful in helping to model select the number of lags to use. First, I ll run the autoregressions manually using the regress command, testing residuals for autocorrelation after each. reg D.gdp L.D.gdp estat bgodfrey reg D.gdp L(1/2).D.gdp estat bgodfrey

4 . reg D.gdp L.D.gdp Source SS df MS Number of obs = 102 F( 1, 100) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = D.gdp Coef. Std. Err. t P> t [95% Conf. Interval] gdp LD _cons estat bgodfrey Breusch-Godfrey LM test for autocorrelation and lags(p) chi2 df Prob > chi reg D.gdp L(1/2).D.gdp H0: no serial correlation Source SS df MS Number of obs = 101 F( 2, 98) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = D.gdp Coef. Std. Err. t P> t [95% Conf. Interval] gdp LD L2D _cons estat bgodfrey Breusch-Godfrey LM test for autocorrelation lags(p) chi2 df Prob > chi H0: no serial correlation I estimated AR(1) and AR(2) models on the differenced series. AR(1) is probably the best choice, but I continue the example with AR(2) just for fun. The arima command is very convenient. It can be used to take differences, add autoregressive terms, add other regressors and their lags, and add autocorrelated errors to the model (called moving average). Here is the syntax:

5 Title [TS] arima ARIMA, ARMAX, and other dynamic regression models Syntax Basic syntax for a regression model with ARMA disturbances arima depvar [indepvars], ar(numlist) ma(numlist) Basic syntax for an ARIMA(p,d,q) model arima depvar, arima(#p,#d,#q) options Model noconstant arima(#p,#d,#q) ar(numlist) ma(numlist) constraints(constraints) collinear description suppress constant term specify ARIMA(p,d,q) model for dependent variable autoregressive terms of the structural model disturbance moving-average terms of the structural model disturbance apply specified linear constraints keep collinear variables I want 2 autoregressive terms and to take the first difference of real GDP. That is done arima gdp, arima(2,1,0). arima gdp, arima(2,1,0) (setting optimization to BHHH) Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = (switching optimization to BFGS) Iteration 5: log likelihood = Iteration 6: log likelihood = Iteration 7: log likelihood = Iteration 8: log likelihood = ARIMA regression Sample: Number of obs = 103 Wald chi2(2) = Log likelihood = Prob > chi2 = OPG D.gdp Coef. Std. Err. z P> z [95% Conf. Interval] gdp ARMA _cons ar L L /sigma The results for the AR terms are very close to those from least squares. ML is not making much of a difference in estimating the parameters. Compare the standard errors though. To generate a series of 1-step ahead forecasts, simply use

6 predict ghat, y Dynamic forecasts can be generated as well. These use actual values of gdp up to a point and then use forecasted values for all subsequent values. These will be quite smooth. predict ghatdy, dynamic(tq(2004q1)) y tsline gdp ghatdy ghat if tin(2004q1,) The resulting graph is q1 2005q3 2007q1 2008q3 2010q1 real US gross domestic product y prediction, one-step y prediction, dyn(tq(2004q1)) You can see that the 1-step forecasts never deviate very far from the actual series (since they use actual values of gdp each time). The dynamic forecast is smoother and deviations of predicted and actual gdp are fairly large (at least for a while).