Stationary Time Series, Conditional Heteroscedasticity, Random Walk, Test for a Unit Root, Endogenity, Causality and IV Estimation

Transcription

1 1 / 67 Stationary Time Series, Conditional Heteroscedasticity, Random Walk, Test for a Unit Root, Endogenity, Causality and IV Estimation Chapter 1 Financial Econometrics Michael Hauser WS18/19

2 2 / 67 Content Notation Conditional heteroscedasticity, GARCH, estimation, forecasting Random walk, test for unit roots Granger causality Assumptions of the regression model Endogeneity, exogeneity Instrumental variable, IV, estimator Testing for exogeneity: Hausman test and Hausman-Wu test Appendix: OLS, Weak stationarity, Wold representation, ACF, AR(1), MA(1), tests for WN, prediction of an AR(1)

3 3 / 67 Notation X, Π... matrices x, β, ɛ... column vectors x, β, ɛ... real values, single variables i, j, t... indices L... lag operator. We do not use the backward shift operator B. WN, RW... white noise, random walk asy... asymptotically df... degrees of freedom e.g.... exempli gratia, for example i.e.... id est, that is i.g.... in general lhs, rhs... left, right hand side rv... random variable wrt... with respect to

4 4 / 67 Conditional heteroscedasticity GARCH models

5 5 / 67 Conditional heteroscedasticity A stylized fact of stock returns and other financial return series are periods of high volatility followed by periods of low volatility. The increase and decrease of this volatility pattern can be captured by GARCH models. The idea is that the approach of new information increases the uncertainty in the market and so the variance. After some while the market participants find to a new consensus and the variance decreases. We assume - for the beginning - the (log)returns, r t, are WN r t = µ + ɛ t with ɛ t N(0, σ 2 t ) σ 2 t = E(ɛ 2 t I t 1 ) σ 2 t is the predictor for ɛ 2 t given the information at period (t 1). σ 2 t is the conditional variance of ɛ 2 t given I t 1. I t 1 = {ɛ 2 t 1, ɛ2 t 2,..., σ2 t 1, σ2 t 2,...}.

6 Conditional heteroscedasticity Volatility is commonly measured either by ɛ 2 t the local variance (rt 2 ɛ t, the modulus of ɛ t. contains the same information), or 6 / 67

7 7 / 67 Generalized autoregressive conditional heteroscedasticity, GARCH The generalized autoregressive conditional heteroscedasticity, GARCH, model of order (1, 1), GARCH(1,1), uses as set of information I t 1 = {ɛ 2 t 1, σ2 t 1 } σ 2 t = a 0 + a 1 ɛ 2 t 1 + b 1σ 2 t 1 The unconditional variance of ɛ t is then [We set E(σ 2 t ) = E(ɛ2 t ) = σ2. So σ 2 = a 0 + a 1 σ 2 + b 1 σ 2 and we solve wrt σ 2.] E(ɛ 2 t ) = a 0 1 (a 1 + b 1 ) The unconditional variance is constant and exists if a 1 + b 1 < 1 Further as σ 2 t is a variance and so positive, a 0, a 1 and b 1 have to be positive. a 0, a 1, b 1 > 0

8 8 / 67 GARCH(r, s) A GARCH model of order (r, s), GARCH(r, s), r 0, s > 0, is σ 2 t = a 0 + s a j ɛ 2 t j + 1 r b j σt j 2 1 with s a j + 1 r b j < 1, a j, b j > 0 1 An autoregressive conditional heteroscedasticity, ARCH, model of order s is a GARCH(0, s). E.g. ARCH(1) is σ 2 t = a 0 + a 1 ɛ 2 t 1 with a 1 < 1, a 0, a 1 > 0 Comment 1: A GARCH(1,0), σt 2 = a 0 + b 1 σt 1 2, is not useful as it describes a deterministic decay once a starting value for some past σt j 2 is given. Comment 2: The order s refers to the number of a j coefficients, r to the b j coefficients.

9 9 / 67 IGARCH(1,1) An empirically relevant version is the integrated GARCH, IGARCH(1,1) model, σ 2 t = a 0 + a 1 ɛ 2 t 1 + b 1σ 2 t 1 with a 1 + b 1 = 1, a 0, a 1, b 1 > 0 where the conditional variance exists, but not the unconditional one.

10 10 / 67 GARCH(1,1): ARMA and ARCH representations We can define an innovation in the variance as ν t = ɛ 2 t E(ɛ 2 t I t 1 ) = ɛ 2 t σ 2 t Replacing σ 2 t by σ 2 t = ɛ 2 t ν t in the GARCH(1,1) model we obtain ɛ 2 t = a 0 + (a 1 + b 1 )ɛ 2 t 1 + ν t b 1 ν t 1 This model is an ARMA(1,1) for ɛ 2 t. So the ACF of r 2 t can be inspected for getting an impression of the dynamics. The model is stationary, if (a 1 + b 1 ) < 1. Recursive substitution of lags of σ 2 t = a 0 + a 1 ɛ 2 t 1 + b 1σ 2 t 1 in σ2 t gives an infinite ARCH representation with a geometric decay σ 2 t = a 0 1 b 1 + a 1 1 b j 1 1 ɛ 2 t j

11 11 / 67 Variants of the GARCH model t-garch: As the unconditional distribution of r t can be seen as a mixture of normal distributions with different variances, GARCH is able to model fat tails. However empirically, the reduction in the kurtosis of r t by a normal GARCH model is often not sufficient. A simple solution is to consider an already fat tailed distribution for ɛ t instead of a normal one. Candidates are e.g. the t-distribution with df > 2, or the GED (generalized error distr) with tail parameter κ > 0. ARCH-in-mean, ARCH-M: If market participants are risk avers, they want a higher average return in uncertain periods than in normal periods. So the mean return should be higher when σt 2 is high. r t = µ 0 + µ 1 σ 2 t + ɛ t with µ 1 > 0, ɛ t N(0, σ 2 t )

12 12 / 67 Variants of the GARCH model asymmetric GARCH, threshold GARCH, GJR model: As the assumption that both good and bad new information has the same absolute (symmetric) effect might not hold. A useful variant is σ 2 t = a 0 + a 1 ɛ 2 t 1 + b 1σ 2 t 1 + γd t 1ɛ 2 t 1 where the dummy variable D indicates a positive shock. So The contribution of ɛ 2 t 1 to σ2 is D t 1 = 1, if ɛ t 1 > 0 D t 1 = 0, if ɛ t 1 0 (a 1 + γ), if ɛ t 1 > 0 a 1, if ɛ t 1 0 If γ < 0, negative shocks have a larger impact on future volatility than positive shocks. (GJR stands for Glosten, Jagannathan, Runkle.)

13 13 / 67 Variants of the GARCH model Exponential GARCH, EGARCH: By modeling the log of the conditional variance, log(σt 2 ), the EGARCH guarantees positive variances, independent on the choice of the parameters. It can be formulated also in an asymmetric way (γ 0). log(σ 2 t ) = a 0 + a 1 ɛ t 1 σ 2 t 1 + b 1 log(σ 2 t 1 ) + γ ɛ t 1 σ 2 t 1 If γ < 0, positive shocks generate less volatility than negative shocks.

14 Estimation of the GARCH(r, s) model Say r t can be described more generally as r t = x tθ + ɛ t with vector x t of some third variables, x t = (x 1t,..., x mt ), possibly including lagged r t s, seasonal dummies, etc. ɛ t with conditional heteroscedasticity can be written as ɛ t = σ t ξ t with ξ t iid N(0, 1) σ 2 t has the form σ 2 t = a 0 + a 1 ɛ 2 t a s ɛ 2 t s + b 1σ 2 t b r σ 2 t r. So the conditional density of r t x t, I t 1 is given by f N (r t x t, I t 1 ) = 1 2πσt 2 exp( 1 2 ɛ 2 t σ 2 t ), t = max(r, s) + 1,..., n Comments: V(σ t ξ t x t, I t 1 ) = σ 2 t V(ξ t x t, I t 1 ) = σ 2 t, starting values σ 2 1,..., σ2 r appropriately chosen, ɛ t = ɛ t (θ), σ 2 t = σ 2 t (a 0, a 1,..., a s, b 1,..., b r ) 14 / 67

15 15 / 67 Estimation of the GARCH model The ML, maximum likelihood, estimator of the parameter vector θ, a 0, a 1,..., a s, b 1,..., b r is given by as the ɛ s are uncorrelated. max θ,a 0,a 1,...,a s,b 1,...,b r n max(r,s)+1 f N (r t x t, I t 1 ) The estimates are asy normal distributed. Standard t-tests, etc. apply. Note the requirement of uncorrelated ɛ s. In case autocorrelation in the returns r t is ignored, the model is misspecified. And, you will detect ARCH effects although they might not exist. So in a first step, model the level of r t by ARMA, e.g., and then fit GARCH models to the residuals. Remark: If r t is autocorrelated, so also r 2 t is autocorrelated i.g.

16 16 / 67 Forecasting a GARCH(1,1) process Using σ 2 = a 0 /(1 (a 1 + b 1 )) in σ 2 t = a 0 + a 1 ɛ 2 t 1 + b 1σ 2 t 1 we rewrite the model as σ 2 t σ 2 = a 1 [ɛ 2 t 1 σ2 ] + b 1 [σ 2 t 1 σ2 ] So the 1-step ahead forecast E(ɛ 2 t+1 I t) is σ 2 t+1 t = (σ2 t+1 ) = E(ɛ2 t+1 I t) = σ 2 + a 1 [ɛ 2 t σ 2 ] + b 1 [σ 2 t σ 2 ] with the 1-step ahead forecast error ɛ 2 t+1 σ2 t+1 t = ν t+1. The h-step ahead forecast is, h 2, [replacing both ɛ 2 t, σ2 t by their (h 1)-step ahead forecast σt+h 1 t 2 ] σt+h t 2 = σ2 + (a 1 + b 1 )[σt+h 1 t 2 σ2 ] = = σ 2 + (a 1 + b 1 ) h 1 [a 1 (ɛ 2 t σ 2 ) + b 1 (σt 2 σ 2 )]

17 17 / 67 Random walk and unit root test Random walk, I(1), Dickey Fuller test

18 18 / 67 Random walk, RW A process y t = αy t 1 + ɛ t with α = 1 is called random walk. y t = y t 1 + ɛ t with ɛ t WN Taking the variances on both sides gives V(y t ) = V(y t 1 ) + σ 2. This has only a solution V(y), if σ 2 = 0. So no unconditional variance of y t exists. Starting the process at t = 0 its explicit form is y t = y 0 + t 1 ɛ j. The conditional expectation and conditional variance are E(y t y 0 ) = y 0, V(y t y 0 ) = t σ 2 The conditional variance of a RW increases with t. The process is nonstationary. The associated characteristic polynomial 1 z = 0 [as (1 L)y t = ɛ t ] has a root on the unit circle, z = 1. Moreover it has a unit root. z = 1

19 19 / 67 Random walks There are 3 standard types of a random walk. ɛ t is WN. Random walk (without a drift): y t = y t 1 + ɛ t Random walk with a drift. c is the drift parameter. y t = c + y t 1 + ɛ t or y t = y 0 + c t + t 1 ɛ j Random walk with drift and trend. y t = c + b t + y t 1 + ɛ t

20 ARIMA(1,1,1) Say we have a process y t = 1.2y t 1 0.2y t 2 + ɛ t 0.5ɛ t 1 written in lag polynomials (1 1.2L + 0.2L 2 )y t = (1 0.5L)ɛ t. The characteristic AR polynomial 1 1.2z + 0.2z 2 = 0 has the roots z 1 = 1/0.2, z 2 = 1. So we can factorize as (1 0.2L)(1 L)y t = (1 0.5L)ɛ t Replacing (1 L)y t by y t = y t y t 1 the process may be written as (1 0.2L) y t = (1 0.5L)ɛ t or y t = 0.2 y t 1 + ɛ t 0.5ɛ t 1 So y t, the differenced y t, is a stationary ARMA(1,1) process. The original process y t (the level) is integrated of order 1, an ARIMA(1,1,1). Taking simple differences removes the nonstationarity and brings us in the world of stationary processes, where elaborate techniques are available. Sometimes differencing twice is necessary. Root for (1 αl): 1 αz = 0 with z 1 = (1/α). So 1 (1/z 1 )z = / 67

21 Test for a unit root, Dickey-Fuller test We start with the model y t = c + αy t 1 + ɛ t, ɛ t WN and want to know whether α = 1, the process is a random walk, or α < 1, the process is stationary. Estimation of α with OLS gives consistent estimates in both cases. For RW s they are even super-consistent, and converge faster than with n. For the Dickey-Fuller test we subtract on both sides y t 1 and get y t = c + (α 1)y t 1 + ɛ t. We estimate with OLS and calculate the standard t-statistic for (α 1) τ = ˆα 1 se(ˆα). 21 / 67

22 22 / 67 Dickey-Fuller test, DF test Under the null hypothesis H 0 : α = 1 or (α 1) = 0 the τ-statistic is distributed according to the distribution tabulated in Dickey-Fuller(1976). It has fatter tails than the t-distribution and is somewhat skewed to the left. The distribution depends on the sample size, and on the type of the RW model, without drift, with drift, with a linear trend: A : y t = y t 1 + ɛ t, B : y t = c + y t 1 + ɛ t, C : y t = c + b t + y t 1 + ɛ t The alternative hypothesis is H A : α < 1 or (α 1) < 0

23 23 / 67 Dickey-Fuller test, deterministic trend Remark: Model C, y t = c + b t + y t 1 + ɛ t, is, used to distinguish between a RW and a deterministic (linear) trend. y t = c + b t + (α 1)y t 1 + ɛ t H 0 : If the DF test leads to an acceptance of the H 0 of a RW, then y t is a RW, and not a deterministic trend. (It might be that the conditional mean has a quadratic trend.) H A : If the DF test rejects the H 0 of a RW, then y t does not obey to a RW. If b is significant, y t might be modeled more adequately by a deterministic trend.

24 24 / 67 The augmented Dickey-Fuller test In case ɛ t is not WN, but (stationary and) autocorrelated, the Dickey-Fuller test is not reliable. Therefore we estimate the model augmented with lagged y t s to capture this autocorrelation, and obtain approximately uncorrelated residuals. y t = c + (α 1)y t 1 + p φ j y t j + ɛ t 1 The maximal lag order p is chosen automatically by an information criterion of your choice, e.g. AIC, SBC. Remark: Tests for bubbles (α > 1) are found in Phillips, Shi and Yu(2015).

25 Granger causality 25 / 67

26 26 / 67 Granger causality Granger causality is actually a concept for classification of predictability in stationary dynamic (AR(p)-type) models. Restricted model, model 0: y t = α 0 + α 1 y t α p y t p + ɛ 0,t We ask whether a stationary x t can help to forecast y t. We consider the model augmented by lagged x s: Unrestricted model, model 1: y t = α 0 + α 1 y t α p y t p + β 1 x t β q x t q + ɛ 1,t X Granger-causes Y, if at least one of the β j 0.

27 27 / 67 Testing via LR-test The hypothesis of no Granger causation of X wrt Y can be tested for fixed p, q via a LR-test: H 0 : β 1 =... = β q = 0 n log( s2 0 s1 2 ) χ 2 (q) The test statistic is distributed χ 2 with q degrees of freedom. log(σ0 2/σ2 1 ) may be interpreted as the strength of the causality.

28 28 / 67 Testing via F-test, instantaneous causality Or we test via a Wald F-statistic (RSS 0 RSS 1 )/q RSS 1 /(n p q) F(q, n p q) If in the augmented model also the current x t plays a role, then this is called instantaneous causality between X and Y. Both Y Granger-causes X and X Granger-causes Y might exist. This is called feedback relationship. [RSS... residual sum of squares]

29 29 / 67 Example, deficiencies Example: Daily returns of various European stock indices were slightly Granger-caused by the New York stock exchange before 9/ However during the months after, adjustment speeded up, instantaneous causality rose and lagged reaction vanished. Granger causation is subject to the same deficiencies as the bivariate correlation coefficient: Causation may vanish when including a third variable (spurious correlation), or Causation may turn out only after adding a third variable. The space within causation is searched for is relevant.

30 Assumptions of classical regression model 30 / 67

31 The linear regression model y = Xβ + ɛ The dependent variable y is (n 1), a column of length n. The regressor matrix X is (n K ), X = [x tk ] n K. The parameter vector β is (K 1) and the error term ɛ is (n 1). More explicitly y = x 1 β x K β K + ɛ x k is the k-th column in X, x k = [x k ] n 1. In terms of single time points or individuals y t = x t β + ɛ t where x t, x i, are the t-th/i-th row in X. 31 / 67

32 32 / 67 Assumptions: review A0. True model: We know the true model and the model in question is specified correctly in the given form. A1. Linear model: The model is linear. (I.e. linear in the parameters.) A2. Full rank of X: No variable can be expressed as linear combination of the others. All parameters are identified. No multicollinearity. As the regressors are stochastic the condition is: plim(x X/n) = Q is finite, constant and non singular. If the x-variables have all mean zero, Q is the asymptotic covariance matrix. In the special case of a deterministic X A2 reduces to the full rank, nonsingularity, respectively, of (X X).

33 33 / 67 Assumptions: review A3. Exogeneity of the explanatory variables: The explanatory variables at any point of time, i, are independent with the current, and all past and future errors. E[ɛ i x j1,..., x jk ] = 0 i, j = 1,..., n. A4. Homoscedasticity and uncorrelated errors: The disturbances ɛ t have finite and constant variance, σ 2. They are uncorrelated with each other. A5. Exogenously generated data: The generating process of the x s is outside the model. The analysis is done conditionally on the observed X. A6. Normal distribution: The disturbances are (approximately) normally distributed (for convenience). y is multivariate normal conditional on X, y X N(.,.)

34 Endogeneity 34 / 67

35 35 / 67 Endogeneity: definition Endogeneity exists if explanatory variables are correlated with the disturbance term. We start with y = Xβ + ɛ and estimate by OLS, with b = ˆβ = (X X) 1 X y. Then b may be written as b = β + (X X) 1 X ɛ If E(X ɛ) or plim(x ɛ/n) is not zero, then the estimates are biased or inconsistent. There are several models where endogeneity arises: The errors-in-the-variables model Simultaneity Dynamic regression

36 36 / 67 Endogeneity: the errors-in-the-(explanotary)variables model The true model is: y = βx + ɛ However, we observe only x o, i.e. x with a measurement (random) error η, which we cannot control for: x o = x + η The model we estimate is: y = βx o + ɛ o So ɛ o = ɛ βη, and Cov(x o, ɛ o ) 0. Cov(x o, ɛ o ) = Cov(x + η, ɛ) β Cov(x + η, η) 0 The first term can be assumed to be zero, Corr(ɛ, η) = 0, the second one is different from zero. The measurement error causes inconsistency of the estimates. The bias in β does not vanish even with n.

37 37 / 67 Endogeneity: simultaneity Simultaneity is a general phenomenon. It says that individual decisions in (economic) systems are interdependent. Single independent decisions can hardly be separated out. Example: In macroeconomics the simple consumption function describes the relationship between private consumption, C, and disposable income, Y. C = α + βy + ɛ However, income is generated by consumption expenditure, and investment and government consumption, etc. Say, the latter is aggregated and denoted by Z. Y = C + Z Both equations give a simultaneous equation system.

38 38 / 67 Endogeneity: simultaneity LS of the consumption function yields (a b) = (α β) + [(1Y ) (1Y )] 1 (1Y ) ɛ Since Y = C + Z, E(1Y ) ɛ = E[1 (C + Z )] ɛ 0 E(C ɛ) 0 as C = α + βy + ɛ, and so (a b) is biased. This is a simple example for the simultaneous equation bias. Example: It is difficult to distinguish empirically between pure univariate expectations and expectations influenced by the overall economic state. Further, consider expectations about the expectations of other subjects,....

39 39 / 67 Strict / weak exogeneity Assumption A3 says that current, past and future disturbances are uncorrelated with current explanatory variables. E[ɛ i x j1,..., x jk ] = 0 i, j = 1,..., n or E[x jk ɛ i ] = 0 i, j = 1,..., n for all k We say the x variables are strictly exogenous. For weak exogeneity only E[x ik ɛ i ] = 0 i = 1,..., n for all k is required.

40 Predetermined variables, dynamic regression Variable x is called predetermined if E[x t ɛ t+s ] = 0 s > 0 Current x t is uncorrelated with all future ɛ s, but possibly correlated with past. Consider the simple dynamic regression with γ < 1 Repeated substitution gives y t = γy t 1 + ɛ t t 1 y t = γ i ɛ t i + γ t y 0 = ɛ t + γɛ t γ t 1 ɛ 1 + γ t y 0 i=0 y t 1 does not dependent on ɛ t, ɛ t+1,.... So y t 1 is predetermined, and weakly exogenous. The lack of contemporary correlation of y t and ɛ t is essential. If E(x t ɛ t ) = 0 the consequences of the dynamic dependencies vanish at least asymptotically and we obtain consistent estimates, which might, however, be biased in small samples. 40 / 67

41 Instrumental variables 41 / 67

42 42 / 67 Instrumental variables, IV A solution for the edogeneity problem (biases/inconsistencies) are IV estimators. Instrumental variables, short instruments, are essentially exogenous variables - they do not correlate with the disturbances by assumption - used as proxies for endogenous explanatories. The art is to find variables which are good proxies (and do not show endogeneity), so that the fit of the regression will become acceptable. Sometimes the loss of information due to the use of IVs (bad proxies) is larger than the bias induced by endogenity. Then it is preferable to stay with the simple LS estimator.

43 43 / 67 IV estimation: a 2-step procedure Say the model is y = X 1 β 1 + X 2 β 2 + ɛ X 1 contains the variables leading to the endogeneity problem. We choose at least as many instruments, the columns in Z, as there are variables in X 1, regress (LS) X 1 on Z : X 1 = Z γ + η. We get X 1 = Z γ, and replace X 1 in the original model by X 1 : y = X 1 β 1 + X 2 β 2 + ɛ The resulting LS estimate b = b IV is the instrumental variable estimator.

44 44 / 67 IV estimator: more formally Formally we relax assumption A3 to AI3 (weak exogeneity) with AI 3. E(ɛ i x i,1,..., x i,k ) = 0, i = 1,..., n Only the contemporaneous correlation has to vanish. We show that the estimate b IV is asymptotically normal. For notational convenience the model is the following y = Xβ + ɛ, X (n K ), Z (n K ) There are - say K - instruments in Z, so that X Z has rank K. LS of X on Z gives with X = Z γ + η γ is (K K ) and η (n K ). γ = (Z Z ) 1 Z X and X = Z γ = Z (Z Z ) 1 Z X

45 45 / 67 IV estimator: more formally Now we have to estimate by LS y = Xβ + ɛ b = b IV = ( X X) 1 X y =... = [X Z (Z Z ) 1 Z X] 1 [Z (Z Z ) 1 Z X] y =... = (Z X) 1 Z y

46 46 / 67 IV estimator: unbiasedness, consistency We replace y by y = Xβ + ɛ. So b IV = β + (Z X) 1 Z ɛ For consistency we need that plim(z Z /n) = Q ZZ plim(z X/n) = Q ZX = Q XZ plim(x X/n) = Q X plim(z ɛ/n) = 0 a positive definite matrix a nonzero matrix a positive definite matrix and as Z are valid instruments. Then plim b IV = β + plim(z X/n) 1 (Z ɛ/n) = β + Q 1 ZX plim(z ɛ/n) plim b IV = β. The IV estimator is consistent. and so

47 47 / 67 IV estimator: asy normality Application of a CLT (central limit theorem) gives the asymptotic distribution of b IV. The expectation is obtained by using (weak) AI3 E(ɛ t z t ) = 0, the variance by considering (Z ɛ/ n). Under very general conditions moments up to order (2 + ɛ) exist, ɛ weakly dependent holds Z ɛ/ n d N[0, σ 2 Q ZZ ] Remark: If the disturbances are heteroscedastic having covariance matrix Ω σ 2 I, the asymptotic variance of (Z ɛ/ n) would be plim(z ΩZ /n) instead. The IV estimator is asymptotically distributed as b IV N[β, (σ 2 /n)q 1 ZX Q ZZ Q 1 XZ ] Remark: asyvar(z ɛ/ n) = plim(z ɛ/ n)(z ɛ/ n) = plim(z ɛɛ Z )/n = σ 2 Q ZZ

48 48 / 67 Hausman test: testing for endogeneity If there is a suspicion of endogeneity about the X s, we want to check for it. The Hausman test offers a possibility. We compare the estimates y = Xb LS + ɛ t,ls and y = Xb IV + ɛ t,iv

49 49 / 67 Hausman test: testing for endogeneity The idea is that under the null hypothesis of no endogeneity both OLS and IV are consistent, though IV is less efficient (because we use proxies as instruments). Asy.Var[b IV ] Asy.Var[b LS ]... nonnegative definite Further, under the null should hold E(d) = E(b IV b LS ) = 0 However under the alternative of endogenity LS is biased/inconsistent, and the estimates differ: E(d) 0. A Wald statistic based thereon is H = d {Est.Asy.Var[d]} 1 d with Est.Asy.Var[d] the estimated asymptotic variance of d.

50 Hausman test: testing for endogeneity Est.Asy.Var[d] depends on the variance of b IV, b LS and their covariances. However, the covariance between an efficient estimator, b E, and its difference from an inefficient one, b I, is zero. (without proof) Cov(b E, b I b E ) = Cov(b E, b I ) Cov(b E, b E ) = 0 So Cov(b E, b I ) = Var(b E ). Applied to b IV and b LS : Asy.Var(b IV b LS ) = Asy.Var(b IV ) Asy.Var(b LS ) This reduces H under the null of no endogeneity to H = 1 σ 2 d [( X X) 1 (X X) 1 ] 1 d χ 2 (K ) where K = K K 0, and K 0 is the number of explanatories which are not under consideration wrt endogeneity. K... no. of possibly endogenous variables. [For y = Xβ + ɛ the variance of b LS is V(b) = σ 2 (X X) 1 ] 50 / 67

51 51 / 67 Hausman test The Hausman test is more generally applicable. If we compare an efficient and an inefficient estimator, which are both consistent, under the null. And, under the alternative the efficient one becomes inconsistent, and the inefficient remains consistent.

52 52 / 67 Hausman-Wu test: testing for endogeneity A simple related test for endogeneity is the Hausman-Wu test. Under the null we consider the model y = Xβ + ɛ Under the alternative we consider the augmented model y = Xβ + X γ + ɛ where X are the K explanatories under suspicion causing endogeneity approximated by their estimates using the instruments. The idea is, that under the null hypothesis of no endogeneity, X represents an irrelevant additional variable, so γ = 0.

53 53 / 67 Hausman-Wu test: testing for endogeneity Under the alternative the null model yields biased estimates. The test statistics is a common F-test with K and (n K K ) degrees of freedom, where the restricted model (γ = 0) is tested against the unrestricted (γ 0) one. Important to note: The test depends essentially on the choice of appropriate instruments.

54 Exercises, references and appendix 54 / 67

55 55 / 67 Exercises Choose 2 out of (2, 3, 4, 5) and 2 of (1G, 6G). 1G Use EViews and Ex1_1_gspc_garch.wf1. Find an appropriate EGARCH model for the return series of the S&P 500. For the period to First test for a unit root in the stock prices, then specify and estimate the ARMA and GARCH models for the returns separately. Compare standard GARCH, GARCH-M, asymmetric GARCH, t-garch and EGARCH. [For estimating ARMA-GARCH models in R see Ex1_gspc_garch_R.txt and Ex1_interest_rates_R.txt. ] 2 Derive (a) the ARMA(1,1) and (b) the ARCH( ) representation of a GARCH(1,1) model.

56 56 / 67 Exercises 3 Has the model y t = 1.7y t 1 0.8y t 2 + ɛ t a unit root? 4 Derive the IV estimator and its asymptotic distribution for homoscedastic and uncorrelated errors. 5 Write down the F-statistic for the Hausman-Wu test. 6G Investigate the type and strength of causality in the US and European stock index returns around 9/11. Test for various types of causality: X causes Y, instantaneous causality, and feedback effects (X causes Y, and Y causes X). Use Ex1_6_Granger_R.txt.

57 57 / 67 References Greene 13.2 Ramanathan 10 Verbeek 8 Phillips, Shi and Jun(2015): Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500, International Economic Review,

58 58 / 67 Appendix Useful formula for OLS White noise, Wold representation, ACF, AR(1), MA(1), lag polynomial, roots

59 59 / 67 Some useful formula for OLS Model y = Xβ + ɛ The normal equations give the 1st order conditions for min β (ɛ ɛ) LS solution for β (X X)b = X y b = (X X) 1 X y In order to see the (potential) unbiasedness of b b = β + (X X) 1 X ɛ The variance-covariance matrix of the estimate with Var(ɛ) = σ 2 I The residual vector e Var(b) = E[(b β)(b β) ] = σ 2 (X X) 1 e = y Xb = y X(X X) 1 X y = [I X(X X) 1 X ]y = My

60 60 / 67 White noise, WN A stochastic process {x t }, t =..., 1, 0, 1,..., is a sequence of rv s, where the rv s have common properties, eg. for strictly stationarity with finite dimensional common distributions. A stochastic process {x t } is a white noise, WN, x t = ɛ t if for x t (and so also for ɛ t ) the (unconditional) mean E(x t ) = E(ɛ t ) = 0, for all t the (unconditional) variance V(x t ) = V(ɛ t ) = σ 2 fixed (independent of t) and the autocorrelations Corr(x t, x t j ) = Corr(ɛ t, ɛ t j ) = 0 for j 0. More general, x t may have a mean, which is different form zero x t = µ + ɛ t

61 Weak stationarity, ACF A process {x t } is weakly stationary, if E(x t ) = µ fixed, independent of t, V(x t ) = γ 0 = σ 2 x fixed, independent of t Cov(x t, x t j ) = γ j independent of t, and depends only on the time distance between the both periods t and (t j), j 0. Weak stationarity refers only to the first two moments of the process. A common distribution is not required. {γ j }, j 0, is called the autocovariance function, ACF, of {x t }. The sample autocovariances are called correlogram. 61 / 67

62 62 / 67 Wold representation theorem Every weakly stationary process may be represented as a (one-sided) weighted sum of a WN process {ɛ t } x t = µ + δ j ɛ t j = µ + ɛ t + δ 1 ɛ t 1 + δ 2 ɛ t with 0 δ2 j <, (square summable) and normalized with δ 0 = 1. As for all t E(ɛ t ) = 0, V(ɛ t ) = σ 2 and Corr(ɛ t, ɛ t j ) = 0 for j 0 E(x t ) = µ + 0 δ je(ɛ t j ) = µ V(x t ) = 0 δ2 j V(ɛ t j ) = σ 2 0 δ2 j < finite, fixed and independent of t. Remark: V(X + Y ) = V(X) + V(Y ), if Corr(X, Y ) = 0.

63 63 / 67 AR(1) An autoregressive process of order 1, AR(1), is given by with ɛ t WN and α < 1. Written as regression model Its Wold representation is Its ACF decays geometrically. (x t µ) α(x t 1 µ) = ɛ t x t = (1 α)µ + αx t 1 + ɛ t = c + αx t 1 + ɛ t x t = µ + α j ɛ t j V(x t ) = γ 0 = σ 2 x = σ 2 Corr(x t, x t j ) = γ j /γ 0 = α j 0 0 α 2 j = σ α 2

64 64 / 67 Forecasting with AR(1) For forecasting we take the conditional expectation of x t based on available information at (t 1), I t 1 = {x t 1, x t 2,..., ɛ t 1, ɛ t 2,...}. The predictor of an AR(1) process x t for period t is E(x t I t 1 ) = (1 α)µ + αx t 1 The useful information reduces to I t 1 = {x t 1 }. Contrary, the unconditional expectation is E(x t ) = µ. The innovation (stochastics) ɛ t which is driving the AR(1) x t = (1 α)µ + αx t 1 + ɛ t is ɛ t = x t E(x t I t 1 )

65 65 / 67 AR(1), roots of the characteristic polynomial The AR(1) can be written by means of the lag polynomial (1 αl) as (1 αl)(x t µ) = ɛ t The roots of the characteristic polynomial are given by 1 αz = 0 with the solution z = 1/α. The stationarity condition α < 1 translates for the root to z = 1/α > 1. The root lies outside the unit circle. This stationarity condition generalizes to AR(p) processes of higher order p. (1 α 1 L... α p L p )(x t µ) = ɛ t 1 α 1 z... α p z p = 0, z > 1 The roots have to lie outside the unit circle.

66 66 / 67 MA(1) A moving average process of order 1, MA(1), is given by (x t µ) = ɛ t + βɛ t 1 with ɛ t WN, and β < 1 for invertibility. (Invertibility guarantees an unique AR representation. MAs are stationary independent of the choice of β.) Its Wold representation is a finite sum of the ɛ s and is the model itself. Its ACF vanishes after lag 1. V(x t ) = γ 0 = σ 2 x = (1 + β 2 )σ 2 Corr(x t, x t 1 ) = γ 1 /γ 0 = β/(1 + β 2 ) < 1/2 Corr(x t, x t j ) = γ j /γ 0 = 0 for j > 1.

67 67 / 67 Tests for white noise The sample autocorrelations are denoted by ˆρ j, ˆρ j = ˆγ j /ˆγ 0. Under WN the sample autocorrelations (for any j > 0) are asy distributed as ˆρ j a N( 1 n, 1 n ) So a 95% coverage interval for ˆρ j under WN is [ 1 n /n, 1 n /n]. A standard test for autocorrelation of zero up to a maximal order m, 1 j m, is the Box-Pierce test. Using a correction for the df s it is called Ljung-Box test. Q = n(n + 2) m 1 ˆρ 2 j n j χ 2 m Under the H 0 : ρ 1 =... = ρ m = 0 Q follows a chi-square distribution with m degrees of freedom.