Kinh tế học - Chapter 5: Applied econometric time series 4th ed. walter enders

Chapter 5Applied Econometric Time Series 4th ed.Walter Enders12An Intervention Model3Consider the model used in Enders, Sandler, and Cauley (1990) to study the impact of metal detector technology on the number of skyjacking incidents: yt = a0 + a1yt–1 + c0zt + et, a1 2003Q4). Under this assumption, the annual growth rate of GDP was estimated to be 2.5% through 2005Q3. Instead, when they set the values of Tj at the 2000Q4 to 2003Q4 period average, the growth rate of GDP was estimated to be zero. Thus, a steady level of terrorism would have cost the Israeli economy all of its real output gains. In actuality, the largest influence of terrorism was found to be on investment. The impact of terrorism on investment was twice as large as the impact on real GDP.24Impulse Responses25Consider a 2-variable model:    The impulse response function is obtained using the moving average representation: Impulse Responses: An Example2627The Residuals vs the Pure Shocks28e1t = (εyt - b12εzt)/(1-b12b21) e2t = (εzt - b21εyt)/(1-b12b21) If we set b12 or b21equal to zero, we can identify the shocksIdentification29e1t = g111t + g122te2t = g211t + g222t or:et = Gt If we let var(1t) = and var(2t) = , it follows that: E1t2t   The problem is to identify the unobserved values of 1t and 2t from the regression residuals e1t and e2t. 30Identification 2If we knew the four values g11, g12 g13 and g14 we could obtain all of the structural shocks for the regression residuals. Of course, we do have some information about the values of the gij. Consider the variance/covariance matrix of the regression residuals: Eee' = Sim’s Recursive Ordering31Sim’s recursive ordering restricts on the primitive system such that the coefficient b21 is equal to zero. Writing (5.17) and (5.18) with the constraint imposed yields yt = b10 – b12zt + g11yt–1 + g12zt–1 + eyt zt = b20 + g21yt–1 + g22zt–1 + ezt  Similarly, we can rewrite the relationship between the pure shocks and the regression residuals given by (5.22) and (5.23) as e1t = yt – b12zt e2t = zt Sims’ Recursive Ordering32e1t = eyt – b12ezte2t = ezt33Hence, it must be the case that: Eetet' = EGtt'G ' Since Eetet' =  and Ett' = , it follows that:In general you must fix (n2 – n)/2 elements for exact identificationHypothesis Tests34Let Su and Sr be the variance/covariance matrices of the unrestricted and restricted systems, respectively. Then, in large samples: (T-c)(log | S r | - log | S u | ) can be compared to a χ2 distribution with degrees of freedom equal to the number of restrictions.Model Selection Criteria35Alternative test criteria are the multivariate generalizations of the AIC and SBC: AIC = T log | S |+ 2 N SBC = T log | S | + N log(T) Where | S | = determinant of the variance/covariance matrix of the residuals and N = total number of parameters estimated in all equations. Granger-Causality36Granger causality: If {yt} does not improve the forecasting performance of {zt}, then {yt} does not Granger-cause {zt}. The practical way to determine Granger causality is to consider whether the lags of one variable enter into the equation for another variable. Block Exogeneity37Block exogeneity restricts all lags of wt in the yt and zt equations to be equal to zero. This cross-equation restriction is properly tested using the likelihood ratio test. Estimate the yt and zt equations using lagged values of {yt}, {zt}, and {wt} and calculate Su. Reestimate excluding the lagged values of {wt} and calculate Sr. Form the likelihood ratio statistic: (T-c)(log | Sr | - log | Su | This statistic has a chi-square distribution with degrees of freedom equal to 2p (since p lagged values of {wt} are excluded from each equation). Here c = 3p + 1 since the unrestricted yt and zt equations contain p lags of {yt}, {zt}, and {wt) plus a constant.To Difference or Not to DifferenceRecall a key finding of Sims, Stock, and Watson (1990): If the coefficient of interest can be written as a coefficient on a stationary variable, then a t-test is appropriate.You can use t-tests or F-tests on the stationary variables.You can perform a lag length test on any variable or any set of variablesGenerally, you cannot use Granger causality tests concerning the effects of a nonstationary variableThe issue of differencing is important. If the VAR can be written entirely in first differences, hypothesis tests can be performed on any equation or any set of equations using t-tests or F-tests. It is possible to write the VAR in first differences if the variables are I(1) and are not cointegrated. If the variables in question are cointegrated, the VAR cannot be written in first differences38If the I(1) variables are not cointegrated and you use levels:Tests lose power because you estimate n2 more parameters (one extra lag of each variable in each equation).For a VAR in levels, tests for Granger causality conducted on the I(1) variables do not have a standard F distribution. If you use first differences, you can use the standard F distribution to test for Granger causality.When the VAR has I(1) variables, the impulse responses at long forecast horizons are inconsistent estimates of the true responses. Since the impulse responses need not decay, any imprecision in the coefficient estimates will have a permanent effect on the impulse responses. If the VAR is estimated in first differences, the impulse responses decay to zero and so the estimated responses are consistent. 39Seemingly Unrelated Regressions40Different lag lengthsyt = a11(1)yt-1 + a11(2)yt-2 + a12zt-1 + e1tzt = a21yt-1 + a22zt-1 + e2tNon-Causalityyt = a11yt-1 + e1tzt = a21yt-1 + a22zt-1 + e2tEffects of a third variableyt = a11yt-1 + a12zt-1 + e1tzt = a21yt-1 + a22zt-1 + a23wt + e2tSims Bernamke42Sims’ Structural VAR43Sims (1986) used a six-variable VAR of quarterly data over the period 1948Q1 to 1979Q3. The variables included in the study are real GNP (y), real business fixed investment (i), the GNP deflator (p), the money supply as measured by M1 (m), unemployment (u), and the treasury bill rate (r). Note that it is Overidentified44rt = 71.20mt + ert (5.59)mt = 0.283yt + 0.224pt – 0.0081rt + emt (5.60)yt = –0.00135rt + 0.132it + eyt (5.61)pt = –0.0010rt + 0.045yt – 0.00364it + ept (5.62)ut = –0.116rt – 20.1yt – 1.48it – 8.98pt + eut (5.63)it = eit (5.64) Sims views (5.59) and (5.60) as money supply and demand functions, respectively. In (5.59), the money supply rises as the interest rate increases. The demand for money in (5.60) is positively related to income and the price level and negatively related to the interest rate. Investment innovations in (5.64) are completely autonomous. Otherwise, Sims sees no reason to restrict the other equations in any particular fashion. For simplicity, he chooses a Choleski-type block structure for GNP, the price level, and the unemployment rate. The impulse response functions appear to be consistent with the notion that money supply shocks affect prices, income, and the interest rate.Blanchard-Quah45Suppose we are interested in decomposing an I(1) sequence, say {yt}, into its temporary and permanent components. Let there be a second variable {zt} that is affected by the same two shocks. The BMA representation is:The Long-run resrtictionAssume that one of the shocks has a temporary effect on the {yt} sequence. It is this dichotomy between temporary and permanent effects that allows for the complete identification of the structural innovations from an estimated VAR. For example, Blanchard and Quah assume that an aggregate demand shock has no long-run effect on real GNP. In the long run, if real GNP is to be unaffected by the demand shock, it must be the case that the cumulated effect of ane1t shock on the Dyt sequence must be equal to zero. Hence, the coefficients c11(k) must be such that4647Since this must be true for all realizationsRecall that:e1t = c11(0)e1t + c12(0)e2t e2t = c21(0)e1t + c22(0)e2tThe four restrictionsRestriction 1: var(e1) = c11(0)2 + c12(0)2 Restriction 2:var(e2) = c21(0)2 + c22(0)2 Restriction 3:Ee1te2t = c11(0)c21(0) + c12(0)c22(0)48Blanchard-Quah49Changes in 1t will have no long-run effect on the {yt} sequence if: Forecast Error Variance Due to Demand-side Shocks50 1 99.0 51.9 4 97.9 80.212 67.6 86.240 39.3 85.6Horizon Output UnemploymentFigure 5.9 Responses of Real and Nominal Exchange Rates 51