VAR Order Selection & Checking the Model Adequacy Mozhgan Raeisian Shima Goudarzi Mike Bronner 17 Jan 2012 Content • Introduction • A Sequence of Tests for Determining the VAR Order o The Impact of the Fitted VAR Order on the Forecast MSE o The Likelihood Ratio Test Statistic o A Testing Scheme for VAR Order Determination • Criteria for VAR Order Selection o Minimizing the Forecast MSE o Consistent Order Selection • Checking the Whiteness of the Residuals • Testing for Nonnormality o Tests for Nonnormality of a Vector White Noise Process o Tests for Nonnormality of a VAR Process • Tests for Structural Change o Chow Tests o Forecast Tests for Structural Change Introduction • Assume a K-dimentional multiple time series 𝑦 , … , 𝑦 with 𝑦 = 1 𝑇 𝑡 (𝑦 , … , 𝑦 ), known to be generated by VAR (p) process, 1𝑡 𝐾𝑡 𝑦 = 𝜈 + 𝐴 𝑌 + … + 𝐴 𝑦 + 𝑢 (1) 𝑡 1 𝑡−1 𝑃 𝑡−𝑝 𝑡 Deriving the properties of the estimators, a number of assumptions were made. As it will rarely be known with certainty whether the conditions hold that are required to derive consistency and asymptotic normality of the estimators, some statistical tools should be used to check the validity of the assumptions. • What to do if the VAR order p is unknown? Choosing p unnecessarily large: 1) will reduce the forecast precision of the corresponding estimated VAR(p) model. 2) also the estimation precision of the impulse response. 3) Approximate MSE matrix will increase with the order p. A Sequence of Tests for Determining the VAR Order • There is not just one correct VAR order for the mentioned process. if (1) is a correct summary of the characteristics of the process 𝑦 , then the same 𝑡 is true for: 𝑦 = 𝜐 + 𝐴 𝑌 + … + 𝐴 𝑦 + 𝐴 𝑦 + 𝑢 𝑡 1 𝑡−1 𝑃 𝑡−𝑝 𝑃+1 𝑡−𝑝−1 𝑡 with 𝐴 = 0. (As according to assumptions, the possibility of zero 𝑃+1 coefficient matrices is not excluded) • It is practical to have a unique number that is called the order of the process. 𝑌 is called a VAR(p) process if 𝐴 ≠ 0 and 𝐴 = 0 for 𝑖 > 𝑝 so 𝑡 𝑃 𝑖 that p is the smallest possible order. This unique number will be called the VAR order. The Impact of the Fitted VAR Order on the Forecast MSE • If 𝑦 is a VAR(p) process, it is useful to fit a 𝑉𝐴𝑅 𝑝 model and not a 𝑡 𝑉𝐴𝑅(𝑝 + 𝑖) because, forecasts from the latter process will be inferior to those based on an estimated VAR(p) model. This result follows from the approximate forecast MSE matrix. Example: ℎ = 1, approximate forecast MSE is: 𝑇+𝐾𝑝+1 (1) = Σ 𝑦 𝑢 𝑇 (1) is an increasing function of the order of the model fitted to the y data. • The approximate MSE matrix is derived from asymptotic theory. Checking whether the result remains true in small samples: • 1000 Gaussian bivariate time series generated with process .02 .5 .1 0 0 𝑦 = + 𝑦 + 𝑦 + 𝑢 , 𝑡 𝑡−1 𝑡−2 𝑡 .03 .4 .5 .25 0 .09 0 𝛴 = 𝑢 0 .04 • 𝑉𝐴𝑅 2 , 𝑉𝐴𝑅 4 , and 𝑉𝐴𝑅 6 fitted to the generated series. • forecasts with the estimated models have been computed. • These forecasts compared to generated post-sample values. • Average squared forecasting errors result for different forecast horizons h and sample size T (Table 1). Table 1. Average squared forecast errors for the estimated bivariate 𝑉𝐴𝑅 2 process based on 1000 realizations Sample Forecast Average squared forecast errors size horizon VAR(2) VAR(4) VAR(6) T h 𝒚 𝒚 𝒚 𝒚 𝒚 𝒚 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 1 .111 .052 .132 .062 .165 .075 30 2 .155 .084 .182 .098 .223 .119 3 .146 .141 .183 .166 .225 .202 1 .108 .043 .119 .048 .129 .054 50 2 .132 .075 .144 .083 .161 .093 3 .142 .120 .150 .130 .168 .145 1 .091 .044 .095 .046 .098 .049 100 2 .120 .064 .125 .067 .130 .069 3 .130 .108 .135 .113 .140 .113 • Forecasts based on estimated 𝑉𝐴𝑅 2 models are clearly superior to the 𝑉𝐴𝑅 4 and 𝑉𝐴𝑅 6 forecasts for sample sizes T = 30, 50, and 100. • Another Example: 1000 three-dimensional time series with the 𝑉𝐴𝑅 1 process .01 .5 0 0 2.25 0 0 𝑦 = .02 + .1 .1 .3 𝑦 + 𝑢 with Σ = 0 1.0 .5 𝑡 𝑡−1 𝑡 𝑢 0 0 .2 .3 0 .5 .74 • 𝑉𝐴𝑅 1 , 𝑉𝐴𝑅 3 , and 𝑉𝐴𝑅 6 models fitted to these data. • Forecasts and forecast errors computed. • Average squared forecast errors are (Table 2). Table 2. Average squared forecast errors for the estimated three-dimensional VAR(1) process based on 1000 realizations Sample Forecast Average squared forecast errors size horizon VAR(2) VAR(4) VAR(6) T h 𝐲 𝐲 𝐲 𝐲 𝐲 𝐲 𝐲 𝐲 𝐲 𝟏 𝟐 𝟑 𝟏 𝟐 𝟑 𝟏 𝟐 𝟑 1 .87 1.14 2.68 1.14 1.52 3.62 2.25 2.78 6.82 30 2 1.09 1.21 3.21 1.44 1.67 4.12 2.54 2.98 7.85 3 1.06 1.31 3.32 1.35 1.58 4.23 2.59 2.79 8.63 1 .81 1.03 2.68 .96 1.22 2.97 1.18 1.53 3.88 50 2 1.01 1.23 2.92 1.20 1.40 3.47 1.48 1.68 4.38 3 1.01 1.29 3.11 1.12 1.44 3.48 1.42 1.77 4.66 1 .73 .93 2.35 .77 1.00 2.62 .86 1.12 2.91 100 2 .94 1.15 2.86 1.00 1.24 3.12 1.12 1.38 3.53 3 .90 1.15 3.02 .93 1.20 3.23 1.03 1.35 3.51 Again forecasts from lower order models are clearly superior to higher order models. Question? • What if the true order is unknown and an upperbound, say M, for the order is known only. Set up a significance test. First, 𝐻 : 𝐴 = 0 0 𝑀 is tested. If this null hypothesis cannot be rejected, we test 𝐻 : 𝐴 = 0 0 𝑀−1 and so on until we can reject a null hypothesis.