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.

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1 US GDP 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. open usa logs gdp scatters gdp diff(gdp) l_gdp diff(l_gdp) --output=display 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. Now test for unit roots using the ADF-GLS test. I chose the constant and trend version, setting the maximum lag to 12 and testing down.

2 adf 12 gdp --test-down gls ct The results are: Augmented Dickey-Fuller (GLS) test for gdp including 11 lags of (1-L)gdp (max was 12) sample size 92 unit-root null hypothesis: a = 1 with constant and trend model: (1-L)y = b0 + b1*t + (a-1)*y(-1) e 1st-order autocorrelation coeff. for e: lagged differences: F(11, 80) = [0.0000] estimated value of (a - 1): test statistic: tau = % 5% 2.5% 1% Critical values: Notice that the test statistic is and is clearly not significant at even 10%. This looks nonstationary. Repeat the analysis on the differenced data. It is possible to remove the trend from the test since differencing should remove any deterministic (linear) trend from the series. Augmented Dickey-Fuller (GLS) test for d_gdp including 10 lags of (1-L)d_gdp (max was 12) sample size 92 unit-root null hypothesis: a = 1 test with constant model: (1-L)y = b0 + (a-1)*y(-1) e 1st-order autocorrelation coeff. for e: lagged differences: F(10, 81) = [0.0633] estimated value of (a - 1): test statistic: tau = asymptotic p-value This is better. The series is stationary in differences. AUTOREGRESSIONS Run the autoregressions using the stationary transformations. You will need to search for the appropriate number of lags to use. For forecasting, I ll use the SC criterion. Test the residuals for candidate model for autocorrelation using LMF or the usual Breusch-Godfrey test. ols d_gdp(0 to -1) const modtest 1 --autocorr

3 ols d_gdp(0 to -2) const modtest 1 --autocorr Model 4: OLS, using observations 1984:3-2009:4 (T = 102) Dependent variable: d_gdp coefficient std. error t-ratio p-value const e-05 *** d_gdp_ e-010 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 100) P-value(F) 4.32e-10 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin's h ? modtest 1 --autocorr --quiet Breusch-Godfrey test for first-order autocorrelation Test statistic: LMF = , with p-value = P(F(1,99) > ) = Alternative statistic: TR^2 = , with p-value = P(Chi-square(1) > ) = Ljung-Box Q' = , with p-value = P(Chi-square(1) > ) = 0.454? ols d_gdp(0 to -2) const Model 5: OLS, using observations 1984:4-2009:4 (T = 101) Dependent variable: d_gdp coefficient std. error t-ratio p-value const *** d_gdp_ e-06 *** d_gdp_ Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(2, 98) P-value(F) 1.98e-09 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn

4 rho Durbin-Watson ? modtest 1 --autocorr --quiet Breusch-Godfrey test for first-order autocorrelation Test statistic: LMF = , with p-value = P(F(1,97) > ) = Alternative statistic: TR^2 = , with p-value = P(Chi-square(1) > ) = 0.52 Ljung-Box Q' = , with p-value = P(Chi-square(1) > ) = The AR(2) has the smaller SC criterion. Also, it s LMF test for serial correlation in the residuals is not significant at 10%. ARIMA SPECIFICATION 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: arima Arguments: p d q [ ; P D Q ] ; depvar [ indepvars ] Options: --verbose (print details of iterations) --vcv (print covariance matrix) --hessian (see below) --opg (see below) --nc (do not include a constant) --conditional (use conditional maximum likelihood) --x-12-arima (use X-12-ARIMA for estimation) --lbfgs (use L-BFGS-B maximizer) --y-diff-only (ARIMAX special, see below) --save-ehat (see below) Examples: arima ; y arima ; y 0 x1 x2 --verbose arima ; ; y --nc If no indepvars list is given, estimates a univariate ARIMA (Autoregressive, Integrated, Moving Average) model. The values p, d and q represent the autoregressive (AR) order, the differencing order, and the moving average (MA) order respectively. These values may be given in numerical form, or as the names of pre-existing scalar variables. A d value of 1, for instance, means that the first difference of the dependent variable should be taken before estimating the ARMA parameters.

5 If you wish to include only specific AR or MA lags in the model (as opposed to all lags up to a given order) you can substitute for p and/or q either (a) the name of a pre-defined matrix containing a set of integer values or (b) an expression such as {1 4}; that is, a set of lags separated by spaces and enclosed in braces. The optional integer values P, D and Q represent the seasonal AR, order for seasonal differencing and seasonal MA order respectively. These are applicable only if the data have a frequency greater than 1 (for example, quarterly or monthly data). These orders must be given in numerical form or as scalar variables. There are actually four ways to get the same result: 1) ols regression of d_gdp onto 2 lags and a constant, 2) an unconditional arima(1,1,0) of gdp, 3) a conditional ARMAX(0,1,0) of gdp with 2 lags of the differences and a constant as regressors, and 4) a conditional arima(2,1,0) of gdp with a constant. The first option is estimated above. The second, arima 0 1 0; gdp(0 to -2) const yields Model 23: ARMAX, using observations 1984:4-2009:4 (T = 101) Estimated using least squares (= MLE) Dependent variable: (1-L) gdp coefficient std. error z p-value const *** gdp_ e-07 *** gdp_ Mean dependent var S.D. dependent var Mean of innovations -6.31e-15 S.D. of innovations Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn The third arima 0 1 0; gdp d_gdp(-1 to -2) const conditional yields Model 24: ARMAX, using observations 1984:4-2009:4 (T = 101) Estimated using least squares (conditional ML) Dependent variable: (1-L) gdp coefficient std. error z p-value

6 const *** d_gdp_ e-07 *** d_gdp_ Mean dependent var S.D. dependent var Mean of innovations -6.82e-15 S.D. of innovations Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn which is exactly the same as what we got above. Admittedly, this is a very clumsy way to estimate this model. I ve used zero AR and MA terms and merely used the arima command to take the difference of gdp. Notice that I also had to add the lagged differences manually. The conditional switch ensures that the estimation is carried out using least squares rather than maximum likelihood. Now, take fuller advantage of the arima command by removing the lagged differences in the regressors and using 2 AR terms in the arima. The respecified model is: arima 2 1 0; gdp const conditional Model 25: ARIMA, using observations 1984:4-2009:4 (T = 101) Estimated using least squares (conditional ML) Dependent variable: (1-L) gdp coefficient std. error z p-value const *** phi_ e-07 *** phi_ Mean dependent var S.D. dependent var Mean of innovations -6.82e-15 S.D. of innovations Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn Real Imaginary Modulus Frequency AR Root Root The results are identical to those above. The advantage to this approach is that you can now see that the model is stable via the root analysis produced by gretl. The moduli are both roots are greater than 1. So, in practice using the last formulation pays some dividends. Also, you can forego the conditional estimation in favor of the possibly more efficient maximum likelihood estimator.

7 FORECASTING There are several types of forecast that you might want to explore. Dynamic and static. Static forecasts are 1-step ahead forecasts based on observed observations that precede the forecasted period. Obviously, one cannot forecast further into the future than a single period (unless you reserve a hold-out sample at the end). Dynamic forecasts take the estimated model and project the estimated relationship into the future, taking forecasts to generate subsequent ones. To generate a series of 1-step ahead static forecasts, try: fcast fit1 smpl gnuplot gdp fit1 --with-lines --time-series --output=display The first line uses the fcast command to compute the forecasts and puts them into a series called fit1. Then I limit the sample from to and plot the actual and forecasted values. The two should track well. Generally, you cannot generate static forecasts beyond without making some assumptions about what the future will be. To forecast this point, add 4 observations to the sample and use fcast again.

8 smpl full dataset addobs 3 arima 2 1 0; gdp const --conditional fcast fit2 --static gdp prediction 2006: : : : : : : : : : : : : : : : : :2 2010:3 The fcast command with the static switch no computes a static forecast for but cannot do so for later periods because the realized value of gdp in is not available. To forecast farther into the future you need to compute dynamic forecasts. First, the plot of the static forecast: smpl gnuplot gdp fit2 --with-lines --time-series --output=display

9 You can see that at there is a 1-step ahead forecast that shows a slight increase in growth. Dynamic forecasting allows us to forecast out-of-sample. Add 6 more observations to the sample and forecast using the dynamic switch. For 95% confidence intervals, z(0.025) = 1.96 gdp prediction std. error 95% interval 2009: : : : : : : : : Forecast evaluation statistics Mean Error Mean Squared Error e+005 Root Mean Squared Error Mean Absolute Error 579 Mean Percentage Error

10 Mean Absolute Percentage Error Theil's U Bias proportion, UM Regression proportion, UR Disturbance proportion, UD The dynamic forecasts appear to be linear (and based on a linear AR model, they should be). SCRIPT open usa logs gdp diff gdp l_gdp scatters gdp diff(gdp) l_gdp diff(l_gdp) --output=display adf 12 gdp --test-down --gls --ct adf 12 diff(gdp) --test-down --gls --c ols d_gdp(0 to -1) const modtest 1 --autocorr --quiet ols d_gdp(0 to -2) const modtest 1 --autocorr --quiet ols d_gdp(0 to -2) const arima 0 1 0; gdp(0 to -2) const

11 arima 0 1 0; gdp d_gdp(-1 to -2) const --conditional arima 2 1 0; gdp const --conditional fcast fit1 smpl gnuplot gdp fit1 --with-lines --time-series --output=display open usa dataset addobs 5 arima 2 1 0; gdp const --conditional fcast --static fcast fit2 --static smpl gnuplot gdp fit2 --with-lines --time-series --output=display open usa arima 2 1 0; gdp const --conditional dataset addobs 6 fcast --dynamic fcast fit3 --dynamic smpl gnuplot gdp fit3 --with-lines --time-series --output=display