Univariate linear models

Transcription

1 Univariate linear models

2 The specification process of an univariate ARIMA model is based on the theoretical properties of the different processes and it is also important the observation and interpretation of the series. Economic theory does not play an important role in the specification of these models. However, conclusions from this type of analysis can be useful to thin in economic terms. For example, the specification of ARIMA models can answer these types of questions: Is there an equilibrium inflation rate? What are the main feautures of the business cycle? How we can project future sales of a firm?

3 3.1. Box-Jenkins methodology Initial specification Unit root tests Analysis of correlograms and partial correlograms Estimation Checking goodness of adjustment Testing hypothesis on the coefficients Residual analysis Comparing with alternative models.

4 When we work with real date, we must take into account that not model is true. Some of the models are useful. This means that the model we will select must fulfill some statistical properties to be correct and, at the same tiem, it must be compared with the alternative models. We must use our knowledge in this comparison. Never propose a model that is not plausible.

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6 Steps in the specification of the model 1. Determine the stationary transformation of the series. 2. Analyze the correlogram and the partial correlogram to determine the most accurate model for the statitonary transformation of the series. 3. Estimate parameters of the model. 4. Diagnosis to check that the model is correctly specified. 5. Based on step 4, propose and estimate alternative structures that can be compared with the initial specification. 6. Once the optimal specification has been chosen, use the model to draw economic conclusions and to forecast.

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8 1.20E E E E E E GDP

9 Trend? Seasonality? Heterogeneous variance?

10 1. Regarding the trend, we observe sistematic growth but with many breaks. - Economic boom with the arrival of democracy. - Economic crisis at the end of the nineties. 2. We observe that fluctuations of the Chilean GDP increment with the level of the series. 3. Seasonality.

11 It seem reasonable to take a logarithmic transformation for mainly two reasons To eliminate conditional heteroskedasticity. To interprect difference of the series as growth rates. Logaritmic transformation do not damage the properties of the series, even when conditional heteroskedasticity is not observed. We only do not take logarithmic transformation of the series when they are already expressed in percentage terms: interest rates, unemployment rates, etc. Important!!! Logarithmic transformation do not eliminate trend properties of the series.

12 LGDP

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14 Series show sistematic growth during the period of analysis. Slow decreasing patter of the correlogram indicates that series is not stationary and needs at least sa regular difference.

15 It is clearly stochastic and thus happen in practically all economic series. If trend were deterministic, it could be eliminated with a constant and a linear trend.

16 Dependent Variable: LGDP Method: Least Squares Date: 08/12/07 Time: 10:16 Sample: 1966Q1 2006Q2 Included observations: 162 Variable Coefficient Std. Error t-statistic Prob. C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) 0

17 If trend were deterministic, residuals of this regression should be stationary.

18 RESID

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20 A more formal test on the necessity to take first differences are the unit root tests. Among them, the most popular is the Dickey- Fuller test.

21 Let s assume a series generated by an AR(1) process y t = c + φy t 1 + a t We test the following hypothesis H 0 : φ = 1 series is not stationary H 1 : φ 1 series is stationary Under the null, we have a random walk that explain processes with local oscilations. We include a constant in the model to compare with the alternative of stationarity. In that case, the mean must be constant and not necessarily zero It is an unilateral test.

22 For numeric reasons, the test is base don the OLS estimation of the following equation y t = c + (φ 1)y t 1 + a t y t = c + φ y t 1 + a t The two hypothesis are now H 0 : φ = 0 H 1 : φ < 0 The t-statistic is obtained in a standard way. However, its asymptotic distribution under the null must be numerically obtained since it does not follow a standard distribution.

23 Test must be completed by Including enough lags. Including deterministic elements such a trend, seasonal dummies, etc. But, critical values of this test can change with the inclusion of deterministic components.

24 a) Sum of residual squares adjusted by the degrees of freedom: T t T t t t c T a v T a k T T S / 1 1 / where: k /T. v v is a punishment parameter. b) AIC (Akaike information criterium) T t t A T a v 1 2 / ) exp(2 c) SIC (Scharz information criterium) T t t v S T a T 1 2 ) ( /

25 Null Hypothesis: LGDP has a unit root Exogenous: Constant, Linear Trend Lag Length: 8 (Automatic based on SIC, MAXLAG=13) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values.

26 The figures, correlograms and unit root tests suggest that series require a difference to become stationary.

27 DLGDP

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29 These results suggest that the series is stationary after 1 regular difference?

30 It seems that there is a nonstationary seasonality. Therefore we need to take a seasonal difference.

31 D1D4LGPD

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33 Correlations are not signifficant for high lags.

34 What suggest the observation of simple and partial correlograms? Regular part seems an AR(1) process and the seasonal part can be interpreted as a MA(4) or AR(4) process. Therefore, we propose the following model ARIMA(1,1,0)xARIMA(0,1,1)

35 ARMA models can be estimated by maximum likelihood. We can obtain this function although observations are not independent.

36 L = f Y T = f(y T /Y T 1 )f(y T 1 ) = f(y T /Y T 1 )f(y T 1 /Y T 2 )f(y T 2 ) = T t=2 f(y t /Y t 1 )f(y 1 ) If we take a logarithm transformation of the likelihood function, we have T logl = log f(y t /Y t 1 ) + log (f y 1 ) t=2 If y t is conditionally normal then its conditional likelihood is defined as f(y t /Y t 1 ) = 1 exp (y t E(y t /Y t 1 )) 2 2πVar (y t /Y t 1 ) 1/2 2Var (y t /Y t 1 ). Besides, if we asume that process is stationary and normal in a way that the marginal distribution of the initial conditions is normal f(y 1 ) = 1 2πσ 2 1/2 exp (y t μ ) 2 2σ 2.

37 Logarithm of the gaussian likelihood is 1 2 T logl = log (f(y t /Y t )) + log (f y 1 ) t=2 T = T 2 log 2π 1 2 T t=2 t=2 log Var(y t /Y t 1 ) (y t E(y t /Y t 1 ) 2 1 Var(y t /Y t 1 ) 2 logσ2 (y 1 μ) 2 2σ 2 In ARMA models, the conditional variance is always constant 1 2 T logl = T 2 log 2π 1 2 (y t E(y t /Y t 1 ) 2 σ2 a t=2 T t=2 log σ a logσ2 (y 1 μ) 2 2σ 2

38 Mean and marginal distribution depends on the model we have. For example, if we asume an AR(1) process μ = c (1 φ) σ 2 = σ a 2 1 φ 2 E(y t /Y t 1 ) = c + φy t 1 Therefore, the gaussian log-likelihood is 1 2 T t=2 logl = T 2 log 2π 1 2 (y t (c + φy t 1 )) 2 (y 1 σ a 2 c (1 φ) )2 2 σ a 2 1 φ 2 T t=2 log σ a logσ a log (1 φ2 )

39 If we consider that inital values are given, then conditional likelihood coincides with the OLS estimator. This is what most of the software packges do: pcgive, E-views, RATS, CATS, etc. If we do not assume that initial observations are constant, maximization problem is nonlinear and requires optimization algorithms to get the optimum. In general, conditional ML is a simple procedure and very adequate if the information provided by the initial values is not very valued.

40 Dependent Variable: D(LGDP,1,4) Method: Least Squares Date: 08/12/07 Time: 11:13 Sample (adjusted): 1967Q3 2006Q2 Included observations: 156 after adjustments Convergence achieved after 8 iterations Backcast: 1966Q3 1967Q2 Variable Coefficient Std. Error t-statistic Prob. AR(1) MA(4) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted AR Roots 0.31 Inverted MA Roots i

41 Test on the parameters of the model Residual analysis Comparison with alternative models. * Following some information criterium. * According to how different models forecast values out of the sample.

42 RESID

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44 Dependent Variable: D(LGDP,1,4) Method: Least Squares Date: 08/12/07 Time: 13:17 Sample (adjusted): 1968Q2 2006Q2 Included observations: 153 after adjustments Convergence achieved after 3 iterations Variable Coefficient Std. Error t-statistic Prob. AR(1) AR(4) R-squared Mean dependent var -7.59E-05 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted AR Roots i i i i

45 Tests on the coefficients: 1. Apply t tests to each of the estimated parameters (if the t-statisc is not significant at the conventional levels simplify the model by droping that variable). 2. Look at the roots of the AR(p) polinomial to see if they are closed to Look at the roots of the polinomials AR(p) and MA(q) to see if we can simplify the system.

46 1) Residual test i) If the model is right, residuals should not have structure. Otherwise, the model ii) can be simplified in the way suggested by the correlogram of the residuals. Test is the average of the residuals is zero. If this average is not zero we should include a constant in the model. H 0 t( ˆ) : 0. ˆ / media desviación T típica We reject H 0 if the absolute value of the t-statistic is higher tan 2. iii) H 0 : a ( k) 0. H Q s 2 Test the absence of autocorrelation in the residuals. If residuals are White noise, then the autorrelations should not be significantly different from zero. Ljung-Box ( Q s 2 2 s 2 ) test 0 : a (1), a (2),..., a ( s 2) 0. If we reject the hypothesis of lack of autocorrelation, there are two options: A) Choose the reasonable doubts we had in the inital specification. B) Specify a model for the residuals and include it in the initial specification

47 1) Test with respect to alternative models even though the initial model has passed all the test on the results of the estimation. That model is only based on one of the possible different specifications. Test alternative models and choose the one that minimizes some information criterium. If you have doubts on the initial model try different alternative specifications or substitute p by p+1, q by q+1. Choose among these alternatives using some information criterium.

48 Dependent Variable: D(LGDP,1,4) Method: Least Squares Date: 08/22/07 Time: 11:27 Sample (adjusted): 1967Q2 2006Q2 Included observations: 157 after adjustments Convergence achieved after 14 iterations Backcast: 1966Q2 1967Q1 Variable Coefficient Std. Error t-statistic Prob. MA(1) MA(4) R-squared Mean dependent var Adjusted R- squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Inverted MA Roots i i -0.86