VAR Order Selection. Umidjon Abdullaev, Ulrich Gunter, Miaomiao Yan. January 16th Vector Autoregressive Models

Transcription

1 Criteria for Vector Autoregressive Models January 16th 2008

2 Criteria for Agenda 1 Introduction

3 Criteria for Agenda 1 Introduction 2 The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination

4 Criteria for Agenda 1 Introduction 2 The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination 3 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC)

5 Criteria for Agenda 1 Introduction 2 The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination 3 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) 4

6 Criteria for Contrary to the simplifying assumption throughout chapters 1 to 3 in Lütkepohl s (2005) textbook, the lag order of a vector autoregressive (VAR)(p) process is usually unknown. However, one should be interested to know the true lag order because it can be shown that the (approximate) mean squared error (MSE) matrix of the one-step ahead predictor increases with lag order p.

7 Criteria for In other words: if the chosen or the assumed lag order is unnecessarily large, the forecast precision of the corresponding VAR(p) model will be reduced. Therefore, it is useful to have certain procedures or criteria for choosing the adequate lag order p.

8 Criteria for In order to do so, there are several possibilities: One could try to visually inspect correlograms (as suggested by Box/Jenkins) similar to the univariate case to determine p. However, this method does not appear to be useful in the multivariate case. An alternative would be to run so-called diagnostic tests, which means that a lag is to be chosen such that the residuals of the process pass diagnostics. But this empirically turned out to be the least reliable approach.

9 Criteria for A more reasonable suggestion is to run statistical tests with the hypothesis that a certain lag equals zero. Some examples for this kind of hypothesis tests would be the likelihood ratio statistic or approximations thereof such as the Wald F statistic or the LM statistic (see e.g. Wooldridge 2006).

10 Criteria for The last and maybe the most elaborated way of coping with the problem of choosing the right lag order is to use information criteria (IC). The common underlying principle of these IC is that each of them minimizes the MSE. Predictors that minimize the MSE are optimal predictors in case the loss of the forecasters can be approximated by squares.

11 Criteria for In the following, we will skip the first and the second method mentioned above as they do not seem to be very accurate and concentrate on the third and the fourth one. Afterwards, we will evaluate the discussed approaches and present a simple empirical example.

12 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Consider the process y t is called a VAR(p) order if y t = ν + A 1 y t A p y t p + u t where A p 0 and A i = 0 for i > p. And p is the VAR order.

13 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination From previous section, we know Σŷ(1) = T + Kp + 1 Σ u T Obviously, Σŷ(1) is an increasing function of the order of the model fitted to the data. Empirically, forecast from lower order models are clearly superior to higher order models.

14 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Assume y t is a stable Gaussian (normally distributed) VAR(p) process, and its log-likelihood function is ln l(β, Σ u ) = KT 2 ln2π T 2 ln Σ u lnl β 1 2 [y (Z I k )β] (I T Σ 1 u )[y (Z I k )β] = (Z Σ 1 u )y (ZZ Σ 1 u )β Equating to zero gives the unrestricted ML estimator β = ((ZZ ) 1 Z I K )y

15 Criteria for Suppose the restrictions on β are Cβ = c The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination where C is a known (N (K 2 p + K)) matrix of rank N and c is a known (N 1) vector. The restricted ML estimator can be found by Lagrangian approach. L β L(β, γ) = lnl(β) + γ (Cβ c) = (Z Σ 1 u )y (ZZ Σ 1 )β + C γ L γ = Cβ c β r = β + [(ZZ ) 1 Σ u ]C [C((ZZ ) 1 Σ u )C ] 1 (c C β) u

16 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination The log-likelihood ratio statistic is λ LR = 2[lnl( δ, Σ u ) lnl( δ r, Σ r u)] where Σ u = 1 T (Y BZ)(Y BZ) Σ r u = 1 T (Y B r Z)(Y B r Z)

17 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Proposition: Let y t be a stationary, stable VAR(p) process with standard white noise u t. (Here, y t is not necessarily Gaussian) β and β r are the (quasi) ML and restricted (quasi) ML estimators. Then the log-likelihood ratio test statistic is λ LR = 2[ln l( δ, Σ u ) ln l( δ r, Σ r u)] = T (ln Σ r u ln Σ u ) (1) = ( β r β) (ZZ (Σ r u) 1 )( β r β) (2) = ( β r β) (ZZ Σ 1 u )( β r β) + o p (1) (3) = (C β c) [C((ZZ ) 1 Σ u )C ] 1 (C β c) + o p (1) (4) = (C β c) [C((ZZ ) 1 Σ r u)c ] 1 (C β c) + o p (1) (5) and λ LR d χ 2 (N)

18 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Proof: To get (1), note that the third item of the log-likelihood function, [y (Z I k )β] (I T Σ 1 u )[y (Z I k )β] = tr[(y BZ) Σ 1 u (Y BZ)] = tr[σ 1 u (Y BZ) (Y BZ) ] }{{} For unbiasedb, is T Σ u lnl( β, Σ u ) = constant T 2 ln Σ u lnl( β r, Σ r u) = constant T 2 ln Σ r u

19 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination To prove (2), by Taylor s theorem (Appendix A.13, Proposition A.3), ln l( β ln l( β) r ) = ln l( β)+ β ( β r β)+ 1 r }{{ 2 ( β r β) 2 ln l( β) } β r β ( β r β) =0 λ LR = 2[ln l( δ, Σ u ) ln l( δ r, Σ r u)] = ( β r β) 2 ln l( β) β r β ( β r β) }{{ } = (ZZ Σ 1 u )

20 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination plim( Σ u Σ r u) = 0 can be used to show (3). To get (4) and (5), from the formula of β r, λ LR = (C β c) [C((ZZ ) 1 Σ u )C ] 1 C((ZZ ) 1 Σ u )((ZZ 1 ) Σ u )((ZZ ) 1 Σ u )C [C((ZZ ) 1 Σ u )C ] 1 (C β c)

21 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination The asymptotic distribution ofλ LR follows Proposition C.15(5): Suppose ˆβ is an estimator of the (K 1) vector β with T ( ˆβ β) d N (0, Σ). If Σ is nonsingular and plim ˆΣ = Σ, then T ( ˆβ β) ˆΣ 1 ( ˆβ β) d χ 2 (K). Here, (ZZ /T ) 1 Σ r u is a consistent estimator of Γ 1 Σ u. Alternatively, one may consider using the statistic in small samples. λ LR /N d F (N, T Kp 1)

22 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Assuming that M is known to be an upper bound for the VAR order, the following sequence of null and alternative hypotheses may be tested using LR tests: H 1 0 : A M = 0 versus H 1 1 : A M 0 H 2 0 : A M 1 = 0 versus H 2 1 : A M 1 0 A M = 0. H0 i : A M i+1 = 0 versus H1 i : A M i+1 0 A M =... = A M i+2 = 0. H M 0 : A 1 = 0 versus H M 1 : A 1 0 A M =... = A 2 = 0 The likelihood ratio statistic for testing the ith null hypothesis is λ LR (i) = T [ln Σ u (M i) ln Σ u (M i + 1) ] d χ 2 (K 2 )

23 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination It is important to realize that the significance levels of individual tests must be distinguished from the Type I error of the whole procedure, because rejection of H0 i means H i+1 0,..., are automatically rejected. Denoting H M 0 D i... H j 0 is rejected in the jth test when it is actually true. ɛ i... the probability of a Type I error for the ith test. ɛ i = Pr(D 1 D 2... D i ) γ j... the significance level of the jth individual test. γ j = Pr(D j )

24 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination It can be shown that for m j and m, j i, λ LR (m) and λ LR (j) are asymptotically independent if H i+1 0,..., H0 i are true. Hence, D m and D j are independent in large samples so that ɛ i = Pr(D 1 D 2... D i 1 ) + Pr(D i ) Pr{(D 1 D 2... D i 1 ) D i } = ɛ i 1 + γ i ɛ i 1 γ i = ɛ i 1 + γ i (1 ɛ i 1 ), i = 2, 3,..., M By induction and ɛ 1 = γ 1 ɛ i = 1 (1 γ 1 )...(1 γ i ), i = 2, 3,..., M.

25 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination For example, take γ i =.05 then ɛ 1 =.05, ɛ 2 = =.0975, ɛ 3 = Hence, the actual rejection probability will become quite substantial if the sequence of null hypothesis to be tested is long.

26 Criteria for The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination Another testing scheme is also suggested in the literature H 1 0 : A M = 0 versus H 1 1 : A M 0. H i 0 : A M =... = A M i+1 = 0 versus H i 1 : A M 0or...or A M i+1 0 Here, H i 0 is tested against the full VAR(M) model. Unfortunately, the LR statistics will not be independent so that the overall significance level is difficult to determine..

27 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) For prediction purposes, it may not be necessary to find the correct lag order p of the data generation process (DGP) as done by running hypothesis tests because one does not automatically obtain a good model for prediction by using hypothesis testing, and hypothesis testing as such can have the backdraw that, with positive probability, one chooses the incorrect lag order even though the time-series length is large.

28 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) In order to maximize forecast precision, one should choose lag order p such that the forecast MSE is minimized. An approximation to the MSE matrix of the one-step ahead predictor is given by the formula: Σŷ(1) = T + Km + 1 Σ u T T denotes the sample size, K the dimension of the time series, m the order of the VAR fitted to the data and Σ u the white-noise covariance matrix.

29 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) However, we do not know Σ u. We have to estimate it: ˆΣ u (m) = T T Km 1 Σ u (m) This is a least squares (LS) estimator of Σ u with degrees of freedom adjustment T /(T Km 1), where Σ u (m) is the maximum likelihood (ML) estimator of Σ u obtained by fitting a VAR(m) model to the data.

30 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) For a better interpretation and for obtaining a unique solution, the final criterion should be a scalar rather than a matrix. We take the determinant and define the resulting criterion as Final Prediction Error (FPE): [ ] T + Km + 1 T FPE(m) = det T T Km 1 Σ u (m) [ ] K T + Km + 1 = det Σ u (m) T Km 1

31 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) Based on the FPE criterion, we choose the estimation ˆp(FPE) of the lag order p such that FPE[ˆp(FPE)] = min {FPE(m) m = 0, 1,..., M} VAR models of order m = 0, 1,..., M are estimated and the corresponding FPE values are computed. Then, the order m minimizing the FPE is chosen as estimate of the true underlying lag order p.

32 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) FPE(m) = [ ] K T + Km + 1 det T Km 1 Σ u (m) The first term in the above equation increases if m increases, whereas the second term decreases for increasing m. Thus, when these two forces are balanced in equilibrium, one obtains the estimate of the lag order.

33 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) Akaike s Information Criterion (AIC) is defined as follows: AIC(m) = ln Σ u (m) + 2mK 2 Σ u (m) is the same as det Σ u (m) and decreasing in m, mk 2 is the number of freely estimated parameters in a VAR model. 2mK 2 /T is the penalty term that increases in m and converges to zero for T. T

34 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) We choose the estimate ˆp(AIC) of p such that AIC(m) is minimized. One could additionally show that FPE(m) and AIC(m) are similar for some constant N, although the AIC has been derived based on a different reasoning. Hence, both criteria have similar properties, for example the following...

35 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) Corollary: The estimators ˆp(FPE) and ˆp(AIC) are not consistent. However, this would be a desirable asymptotic property of an estimator. An estimator ˆp is consistent if: lim Pr{ˆp = p} = 1 T An estimator ˆp is strongly consistent if: Pr{lim ˆp = p} = 1

36 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) A criterion Cr(m) = ln Σ u (m) + mc T T with c T denoting a non-decreasing sequence of real numbers which depends on the sample size T and maximum lag order M (M p) is supposed to be minimized by choosing ˆp. Then ˆp is a consistent estimator if and only if c T and c T /T 0 for T.

37 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) ˆp is strongly consistent if and only if the above holds and, in addition, c T /2 ln ln T > 1 for T. If M > p, ˆp(AIC) and due to similarity also ˆp(FPE) are not consistent estimators because in this case mc T /T = 2mK 2 /T or c T = 2K 2, which contradicts c T and c T /T 0 for T. Both estimators usually overestimate the true lag order with positive probability.

38 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) The Hannan-Quinn Criterion (HQ) is defined as follows: HQ(m) = ln Σ u (m) + 2 ln ln T T mk 2 We choose the estimate ˆp(HQ) such that HQ(m) is minimized.

39 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) Here c T = 2K 2 ln ln T. c T and c T /T 0 for T. ˆp(HQ) is consistent. ˆp(HQ) is even strongly consistent for K > 1 because in this case c T /2 ln ln T = K 2 > 1.

40 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) The Schwarz Criterion (SC) is defined as follows: SC(m) = ln Σ u (m) + ln T T mk 2 We choose the estimate ˆp(SC) such that SC(m) is minimized.

41 Criteria for Minimizing the Forecast MSE Akaike s Information Criterion (AIC) Hannan-Quinn Criterion (HQ) Schwarz Criterion (SC) Here c T = K 2 ln T. c T and c T /T 0 for T. Moreover, c T /2 ln ln T = K 2 ln T /2 ln ln T > 1 for any K. ˆp(SC) is always strongly consistent.

42 Criteria for Small sample comparison of AIC, HQ, and SC Proposition 4.3. Let y M+1,..., y 0, y 1,..., y T be any K-dimensional multiple time series and suppose that VAR(m) models, m = 0, 1,..., M, are fitted to y 1,..., y T. Then the following relations hold: ˆp(SC) ˆp(AIC) ift 8 ˆp(SC) ˆp(HQ) for allt ˆp(HQ) ˆp(AIC) ift 16.

43 Criteria for A consequence of previous results is that lim T Pr{ˆp(AIC) < p}= 0 and lim Pr{ˆp(AIC) > p}> 0 The same holds for FPE because it is asymptotically equivalent to AIC. Large sample results sometimes are good approximations only if extreme sample sizes are available.

44 Criteria for A sequential testing procedure may give the same order as a selection criterion if the significance levels are chosen accordingly. For instance, AIC chooses an order smaller than the maximum order M if AIC(M 1) AIC(M) or, equivalently, if λ LR (1) = T (ln Σ u (M 1) ln Σ u (M) ) < 2MK 2 2(M 1)K 2 For K=2, 2K 2 = 8 χ 2 (4).90.

45 Criteria for The sequential testing procedure will not lead to a consistent order estimator if the sequence of individual significance levels is held constant. For M > p and a fixed significance level γ, is rejected with probability γ. H 0 : A M = 0 This problem can be circumvented be letting the significance level go to zero as T.

46 Criteria for Some simulation results Data: quarterly, seasonally adjusted series for fixed investment, disposable income, consumption expenditures for West Germany in billions of DM, 1960Q1-1982Q4;