Advanced Econometrics

Transcription

1 Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 2, 2013

2 Outline Univariate time series Multivariate time series General concepts Cointegration Vector autoregressions Multivariate cointegration Panels

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4 General concepts Issues with multivariate time series Regressing integrated variables on other integrated variables often yields apparently significant coefficients (even if there is no relation) and integrated residuals: spurious regression; If residuals are stationary, the relation may be an important dynamic equilibrium: cointegration; Cointegration is closely related to the dynamic mechanism that is directed toward the equilibrium: error correction; The multivariate generalization of the autoregressive model is the vector autoregression (VAR), which sets a convenient frame for cointegration, causality etc.

5 General concepts A simple dynamic model Consider two stationary variables Y and X, with X regarded as exogenous, related by an autoregressive distributed lags (ARDL) model Y t = δ +θy t 1 +φ 0 X t +φ 1 X t 1 +ε t, with white-noise ε t and θ < 1. The entity is called the impact multiplier. Y t X t = φ 0

6 General concepts The equilibrium multiplier Straightforward evaluation yields Y t+1 X t = θφ 0 +φ 1 and Y t+2 X t = θ Y t+1 X t = θ(θφ 0 +φ 1 ) The sum of all these multipliers describes the long-run effect j=0 Y t+j = φ 0 + θφ 0 +φ 1 X t 1 θ = φ 0 +φ 1 1 θ, the long-run multiplier or equilibrium multiplier β.

7 General concepts Error correction X and Y are stationary, for this reason E(Y t ) = δ 1 θ +βe(x t) = α+βe(x t ) for some constant α. Some manipulation yields Y t = φ 0 X t (1 θ)(y t 1 α βx t 1 )+ε t, an error-correction model: Y adjusts to deviations from equilibrium, with 1 θ representing the intensity of adjustment.

8 General concepts Generalizations to higher lag orders Assume the general form θ(l)y t = δ +φ(l)x t +ε t, with lag polynomials of orders p and q and invertible θ(z). Applying the inverse operator θ 1 (L) yields Y t = θ 1 (L)φ(L)X t +θ 1 (L)ε t, and the long-run multiplier becomes θ 1 (1)φ(1) = φ 0 +φ φ q 1 θ 1... θ p, with the denominator positive because of stability.

9 Cointegration The basic experiment of spurious regression Presume X t and Y t are independent random walks. The regression Y t = α+βx t +error tends to produce significant coefficients and non-zero R 2. Flanking diagnostics such as the Durbin-Watson statistic tend to indicate that something is wrong. In fact, the residual is I(1) by construction. To avoid spurious regression, first subject Y and X to unit-root tests. If these tests support the null of I(1), test residuals. If unit-root tests on residuals fail to reject, we have a spurious regression problem. Note, however, that the usual Dickey-Fuller significance points are invalid for residual unit-root tests.

10 Cointegration Non-spurious regression with integrated variables Consider two I(1) variables Y t and X t, with Y t βx t stationary I(0). Then, Y and X are said to be cointegrated. The regression Y t = α+βx t +error yields a coefficient estimate ˆβ that is super-consistent for β, i.e. T(ˆβ β) 0, T(ˆβ β) D, with a non-standard limit distribution D. Note that errors are I(0) but they are usually not white noise here. This regression is called a cointegrating regression.

11 Cointegration Cointegration with two variables Definition If Y t and X t are both I(1) and there exists a β such that Z t = Y t βx t is I(0), then X and Y are cointegrated and (1, β) is the cointegrating vector. The cointegrating relation can be interpreted as a long-run equilibrium in a non-stationary dynamic model, and the model can also be written as an error-correction model (see below).

12 Cointegration Testing for cointegration in a simple regression In order to determine whether the regression is spurious or cointegrating, it is recommended to first test X and Y for unit roots. If the unit-root tests fail to reject, run the regression (no lags!) and apply unit-root tests to residuals. Use different significance points from the usual DF test. Appropriate significance points have been calculated by Phillips & Ouliaris, for example. If the residual unit-root test rejects, there is evidence on cointegration; If the residual unit-root test fails to reject, the regression is spurious.

13 Cointegration Cointegration and error correction Engle & Granger have shown that, if X and Y are cointegrated, then there exists an error-correction representation θ(l) Y t = δ +φ(l) X t 1 γz t 1 +α(l)ε t, with Z t = Y t βx t. If Y deviates positively from the equilibrium at t 1, Y corrects back to it at t by being smaller than usual, and similarly for negative deviations. X could also adjust, but this can only be captured in a full multivariate model.

14 Vector autoregressions Vector autoregressions: the concept The vector autoregressive model (VAR) is a multivariate generalization of the univariate AR model: variables are vectors, coefficients are matrices: Y t = δ +Θ 1 Y t Θ p Y t p +ε t, with Y t,y t 1,...,Y t p,ε t,δ k vectors and Θ j,j = 1,...,p k k matrices. The lag polynomial is now a matrix polynomial Θ(L) = I k Θ 1 L... Θ p L p, with I k denoting the k dimensional identity matrix. In short, we write Θ(L)Y t = ε t.

15 Vector autoregressions Stationarity of multivariate processes In analogy to the univariate case, the vector variable Y t is said to be (covariance) stationary, iff EY t = µ = (µ 1,...,µ k ) ; E(Y t µ)(y t µ) = vary t = Σ Y ; E(Y t µ)(y t h µ) = cov(y t,y t h ) = Γ(h). The matrix-valued function Γ(h) is the cross-covariance function, its standardized version is the cross-correlation function (CCF). It is skew symmetric, i.e. Γ(h) = Γ( h). Also note that Γ(0) = Σ Y.

16 Vector autoregressions Multivariate white noise A k dimensional variable (ε t ) is said to be white noise, iff ε t is stationary; Eε t = (0,...,0) ; Γ(h) = 0 for h 0. Note that component j is uncorrelated with k at all leads and lags, but typically correlated simultaneously, i.e. Σ ε is not usually diagonal.

17 Vector autoregressions Stability of the VAR O.c.s. that the VAR is stable iff all solutions (roots) ζ of the (scalar) determinantal polynomial equation detθ(l) = 0 fulfill the condition ζ > 1. Then, it makes sense to consider the infinite-order MA representation Y t = Θ(1) 1 δ +Θ(L) 1 ε t = µ+a(l)ε t.

18 Vector autoregressions The impulse response function The coefficient matrices of the MA representation of a VAR can be interpreted as derivatives A h = Y t+h ε, t and thus as indicating the response in the components of Y to shocks in the components of the error process ε. For this reason, the k 2 components of the function A(h) are called the impulse-response function. A(h) does not really indicate the response to shocks in the components of past variables Y, due to dynamic feedback. A(h) also fails to reflect the immediate response of other components in ε, as Σ ε is not generally diagonal.

19 Multivariate cointegration Multivariate cointegration: the concept Among k I(1) variables, there may be up to k 1 linearly independent cointegrating vectors. The cointegrating rank r may be any value in {0,1,...,k 1}. Regression methods are an unreliable device for the estimation of the cointegrating vectors that may be summarized in a k r dimensional cointegrating matrix of rank r, whose columns are cointegrating vectors. Multivariate cointegration is best described in the framework of a vector autoregression. Assume that all components of the VAR are either I(0) or I(1).

20 Multivariate cointegration Error-correction representation of VAR Every VAR of order p Y t = δ +Θ 1 Y t Θ p Y t p +ε t can be re-written as a vector error-correction model (VECM) Y t = δ +Γ 1 Y t Γ p 1 Y t p+1 +ΠY t 1 +ε t, with matrices Γ j and Π being functions of Θ 1,...,Θ p, in particular Π = I k +Θ Θ p = Θ(1) for the long-run matrix Π, which represents all cointegration features in the system.

21 Multivariate cointegration The long-run matrix Π All terms in the model Y t j,j = 0,...,p 1, ε t,δ are stationary, so ΠY t 1 must also be stationary. Three cases are of interest: Π = 0 and rkπ = 0, the model is a VAR in Y, and there is no cointegration (k unit roots); rkπ = k, any vector β yields β Y stationary, hence Y is already I(0) and stationary (0 unit roots); 0 < rkπ = r < k, we can represent Π = γβ with k r matrices γ and β (k r unit roots). r is the cointegrating rank, and all columns of β form a basis of the cointegrating space. Π and r are unique, though α and β are not.

22 Multivariate cointegration No drift in the system Take expectations on the VECM form ( Y is stationary): (I Γ 1... Γ p )E Y t = δ +γe(β Y t 1 )+Eε t If there is no linear trend in any Y component, the l.h.s. is 0, and so is the r.h.s., which implies that δ is proportional to γ or δ = γα, which yields the VECM with restricted intercepts Y t = γ( α+β Y t 1 )+Γ 1 Y t Γ p 1 Y t p+1 +ε t.

23 Multivariate cointegration Example: cointegration in a bivariate VAR Consider ( Yt X t ) ( θ11 θ = 12 θ 21 θ 22 )( Yt 1 X t 1 ) ( ε1.t + ε 2.t ), with ( θ11 1 θ Π = Θ(1) = 12 θ 22 1 Cointegration implies that rkπ = 1 and that the determinant is 0, i.e. (θ 11 1)(θ 22 1) θ 12 θ 21 = 0 or Π = θ 21 ( θ11 1 θ 12 θ 21 θ 12 θ 21 θ 11 1 ). ).

24 Multivariate cointegration Example: the bivariate VECM For the bivariate VECM, we obtain ( ) ( ) Yt 1 = θ X 21 {(θ 11 1)Y t 1 +θ 12 X t 1 }+ t θ 11 1 ( ε1.t ε 2.t ), with the recognizable cointegrating vector (θ 11 1,θ 12 ) = β. One can check that (θ 11 1)Y t +θ 12 X t is stationary. Note that γ = (θ 11 1,θ 21 ) θ and β = (1, 12 θ 11 1 ) would also work.

25 Multivariate cointegration Testing for cointegration in a VAR For k > 2, regression-based testing for cointegration using residuals from cointegrating regressions becomes unreliable. The likelihood-ratio test statistic on the cointegrating rank H 0 : r r 0 vs. H A : r > r 0 uses the smallest canonical correlations between variables and their first differences (eigenvalues of specific matrices) ˆλ 1 ˆλ 2... ˆλ k with 0 < ˆλ j < 1. The trace statistic λ(r 0 ) = T k j=r 0 +1 log(1 ˆλ j ) has a non-standard distribution under H 0. Tabulated significance points must be used. These differ for the case of a no-drift restriction.

26 Multivariate cointegration Recommended steps for the Johansen procedure 1. Make sure that all component variables are either I(0) or I(1). The procedure works if some variables are stationary. 2. Determine a lag order p for the VAR, e.g. by AIC. 3. Decide whether to use the no-trend restriction. 4. Determine the cointegrating rank by the trace test sequence, using a VAR with p lags. 5. Use the estimates for all coefficients from a VECM estimation that uses the rank r and p 1 augmenting lags. This step yields ˆγ and the cointegrating vector(s) ˆβ, among others. Why not estimate the cointegrating vector from a regression after step 4? This is much less efficient.

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