ECO400 – PART I: TIME SERIES

ECO400 – PART I: TIME SERIES Numan ÜLKÜ, Ph.D. Associate Professor of Economics and Finance

Types of Data • Cross-Sectional: observations from many units for one single period. Denoted: Xi e.g. 2010 annual inflation figures for 80 countries • Time Series: observations for the same unit over many periods. Denoted: Xt e.g. annual inflation figures for Bulgaria from 1980 to 2010 • Panel Data: combination of time-series and cross section. Denoted: Xi,t e.g. annual inflation figures for 80 countries from 1980 to 2010.

Multiple Regression Question: ISEt = β0 + β1BUXt + εtSuppose we find a significant positive relationship (β1) between ISE and BUX. However, it is well known that both Turkish and Hungarian stock market follows (i.e. moves together with) world markets. Does the significant β1 reflect an exclusive relationship between BUX and ISE, or it is just an artifact of both BUX and ISE moving together with world markets ?

To answer this question, we have to add world market returns into the regression. ISEt = β0 + β1BUXt + β2Wt + εt This is called “controlling for W”. If β1 is still significant , then we can say there is a relationship between BUX and ISE. Otherwise, the apparent relationship between BUX and ISE arises because both BUX and ISE are correlated with W. Failing to include any significant variable in the regression equation would amount to misspecification (omitted variable bias). (principle of generosity vs. principle of parsimony)

Multiple Regression: Yt = β0+ β1X1t + β2X2t + … + βkXkt + εt X2t = θ0+ θ1X1t + θ3X3t + … + θkXkt + vt Yt = λ0 + λvt + et λ = β2is called partial regression coefficient

Lags of a Variable: Xt-1 , Xt-2 , Xt-3 , …. , Xt-k Auto-regression: Xt = β1Xt-1 + β2Xt-2 + β3Xt-3 + …. +βkXt-k is called k-th order autoregressive AR(k) model. (no constant term is included)

A Time-Series Econometrician has to deal with three issues: • Non-stationarity: variables with non-constant mean • Persistence: variables that are serially correlated • Endogeneity: bilateral dynamic interaction between variables

STATIONARITY • A variable is called stationary if it displays mean-reverting behavior. • Any regression with nonstationary variables is invalid. • Hence, any time-series application must start with two preliminary steps: • Test stationarity of the variables • If they are not, convert them into a stationary form

A regression with nonstationary variables will typically reveal the problem with a Durbin-Watson (DW) statistic being significantly smaller than 2. DW statistic measures the first-order autocorrelation in the error term. DW << 2 implies positive autocorrelation in the error term. -------------------- Financial Markets Application: All price series are typically nonstationary. Therefore, we use returns. Rt = ln(Pt / Pt-1)

TESTING STATIONARITY: UNIT ROOT TESTS ADF Test and PP Test: H0 : the series is non-stationary (i.e. it has a unit root) ADF and PP test statistics need to be compared to McKinnon critical values. If H0 can be rejected (the test statistic more negative than the critical value), then the variable can be used in regression.

ADF: Δyt = α + βt + ψyt-1 + Δyt-1 +…….+ Δyt-p +εt PP: Δyt = ψyt-1 + ut H0 : ψ = 0 HA: ψ < 0 (PP test makes a nonparametric adjustment for lagged changes.) Test equation derived from the primitive form: Yt = βYt-1 + et β < 1  stationary ; β = 1  non-stationary ; β > 1  explosive KPSS Test: H0: the series is stationary HA: the series is non-stationary

Treating Non-stationary variables • Before using a non-stationary series in any regression, we have to first treat it. Possible Remedies: 1) first-difference it: Δyt = yt− yt-1 A series is: I(0) if it is stationary I(1) if it becomes stationary when differenced once I(2) if it becomes stationary when differenced twice 2) adjust for trend Sometimes a series can become stationary after de-trending, the it is called trend-stationary. 3) Inflation adjustment: sometimes works, always useful

A variable Xtiscalled white noise if: Xt~ i.i.d. (0, 2) When Yt and Xt are both white noise, then a regression analysis in the form Yt = β0+β1Xt+εt is adequate, otherwise the problem of serial correlation in residuals will arise. Portmanteau Test: H0: Xt is white noise HA: Xt is not white noise

Summary: Preliminary Checks 1) Plot the time-series (form an idea about the nature of the series: is it stationary, does it have non-zero mean and trend term?), check for any outliers (regression results are very sensitive to outliers, so one should investigate the reason, and check for any errors). 2) Perform unit root tests (ADF, KPSS). 3) Perform white noise tests (this will give some idea about the appropriate model). 4) Check normal distribution (Jarque-Bera test).

Univariate Time-Series Modeling: ARMA models Univariate models are used to model and predict the time-series behavior of a variable using information contained in its past values only. They are a-theoretical. ARMA(p,q) Model: yt = μ + ρ1yt-1 + ρ2yt-2 +…+ ρpyt-p + θ1εt-1 + θ2εt-2 +…+ θqεt-q + εt ARIMA model is an ARMA model for a non-stationary variable. An ARIMA(p,d,q) model for an I(d) variable is equivalent to an ARMA(p,q) model in the variable differenced d times. ** AR(p) or ARMA(p,q) models can be used to identify shocks (innovations) of a variable.

ENDOGENEITYandVARMODELING • Most of the economic variables do both affect and affected by each other, both simultaneously and at lags. VECTOR AUTOREGRESSION (VAR) • A bivariate VAR(1) Model Ft= β1,1Ft-1 + β1,2Rt-1 + e1,t Rt=β2,1Ft-1 +β2,2Rt-1+e2,t • Lagged values are predetermined hence lag coefficients can be identified. However, simultaneous coefficients cannot be identified without identifying assumptions (restrictions).

Impulse Response Functions (IRFs) • Are an efficient tool to convey the full information that can be derived from a VAR estimation. They show the response at each period along with the net cumulative response. • Can be cumulative or non-cumulative • We track the response to a 1- SD shock in the impulse variable. Shock means an unexpected change (innovation) in the impulse variable. E.g. R F means we are tracking the response of F to a 1-SD e2,t

Identification of contemporaneous coefficients • Contemporaneous coefficients in a VAR system cannot be identified without identifying restrictions. They form a simultaneous equation system with more unknowns than available equations. • The typical solution is to impose a Cholesky ordering. (The ordering is based on intuition). • Other solutions include Generalized IR (Pesaran and Shin (1998).

Variance Decompositions • In order to measure the explanatory power of all endogenous variablesin the VAR system in explaining the each one of the variables in the VAR system, we use Variance Decompositions (innovation accounting). • It is called Forecast Error Variance Decomposition (FEVD). We can compute the percentage of the forecast error variance of each variable in the system accounted for by each of the all variables in the system, cumulatively. • Obviously, for the contemporaneous period (horizon 1), FEVD only reflects the assumption imposed.

Granger causality Suppose a bivariate (Y and Z) model of lag order 2. Yt = β1Yt-1+β2Yt-2+β3Zt-1+β4Zt-2+ε1,t H0: β3=β4=0 ≡ Z does not Granger-cause Y Zt = ψ1Yt-1+ψ2Yt-2+ψ3Zt-1+ψ4Zt-2+ε2,t H0: ψ1=ψ2=0 ≡ Y does not Granger-cause Z H0 is tested via a Wald test, compared against 2 distribution or via an F-test.

Generalized IRF (Peseran and Shin, 1998) Although Peseran and Shin’s derivation is quite complex, it can be shown that Generalized IRF is equivalent to ordering the shock variable the first in Cholesky factorization. Thus, while JMulti does not have a Generalized IRF feature, it is easy to obtain Generalied IRFs by simply alternating the Cholesky ordering.

COINTEGRATION The variables in the VAR system may have a long-run equilibrium relationship (kind of a parity between them) to which any deviating variable is gradually pulled over time. The long-reun equilibrium relationship is called the cointegrating vector. When there is a significant cointegrating vector, the VAR model should be augmented with an Error Correction term. In other words, pure VAR can be applied only when there is no cointegrating relationship among the variables in the VAR system.

Hence, a prerequisite before running any VAR model is to run a cointegration test. General Reminder: Before running a VAR model, one has to perform two preliminary checks: • 1) Make sure all variables entering the VAR system are stationary by running unit root tests (such as ADF). • 2) Check for any cointegrating relationship among the variables in the system. Variables in a cointegration test have to be I(1), or more generally, integrated of the same order k, where k>0.

COINTEGRATION TESTS A cointegrating relationship signifies a stable linear equation among the variables in levels, whose error term is stationary. Thus, it can be tested in 2 steps: In Step 1, run a regression with I(1) variables . In Step 2, apply unit root test to the error term of this regression. [Engle and Granger, 1987] However, Johansen (1995) cointegration test has been more popular. In these tests, Max. Eigenvalue statistic or Trace statistic is compared to special critical values.

Johansen Cointegration Test H0 : There is 0 cointegrating vector HA : There is 1 cointegrating vector If H0 is rejected, this procedure is repeated sequentially for higher numbers of cointegrating vectors: H0 : There is k cointegrating vectors HA : There are k+1 cointegrating vectors ........... until H0 fails to be rejected. When “ H0 : There is k cointegrating vector ” cannot be rejected, then the number of cointegrating vectors is k.

VECM Models yt = (′yt-1) + A1yt-1 + A2yt-2 +….+ Apyt-p + εt For example: A bivariate VECM of lag order 1: y1 = 1(y1,t-1–y2,t-1) + a1,1y1,t-1 + a1,2y2,t-1 + ε1,t y2 = 2(y1,t-1 –y2,t-1) + a2,1y1,t-1 + a2,2y2,t-1 + ε1,t ′yt-1is the cointegrating equation  are error correction coefficients ′yt-1 = 0 is the EC term. Note: Relationship can be expressed in the form of [ f(y,x,z)=0 ] or [ y=g(x,z) ].

Cointegration framework has 2 major uses: 1) Specifying the long-run equilibrium relationship - the cointegrating vector- 2) Generalizing a short-run dynamic model by including an EC term. - VECM – Other uses: • Can measure the speed of adjustment ()

Restrictions can be imposed on the cointegrating vector: i = 0 implies that variable i is weakly exogenous (i.e.it does not “correct”) i = 0 implies that variable i is not part of the cointegrating relationship. Note: JMulti normalizes the cointegrating equationsuch that 1 = 1. Then, it is important that variable 1 (i.e. the variable which is placed first in the ordering) significantly enters the cointegrating equation. Otherwise, the coefficients of the reduced form cointegrating equation are incorrect.

Issues in cointegration not satisfactorily resolved yet:1) Sensitivity of results to different specifications and sometimes even orderings2) Difficulty in assigning meaningful interpretations to long-run cointegrating equation3) Parameters of the cointegrating equation are not uniquely identified, hence results are sensitive to the ordering of the variables4) Dynamic lags and EC term may substitute each other

Student’s Guide on Cointegration Applications • Preparation: Make sure data is in log-levels, and inflation-adjusted (as levels may drastically change as a result of cumulative inflation over long periods). Perform the unit root test, and make sure all variables are I(1) or higher same order. • Cointegration test: This is a crucial preliminary step, because even if there is no real (significant) cointegrating vector, the algorithm will estimate one. • To make sure you obtain a correct normalization of the cointegrating equation, make sure you order first a variable which participates in the cointegrating equation. • Obtaining the correct cointegration equation can always be problematic when the k>2. Our practical solution shall be to apply the (initial) cointegration test to all possible bivariate combinations, and then chose the variable that enters the most cointegrating relationships (with all of the other variables). IRFs however will be reliable in any case.

Learning Objectives: The student is expected to: 1) be able to perform a full cointegration-VECM analysis when k=2. 2) be aware that cointegrating vector is not uniquely identified, and question (whenever k>2 in any real-life exercise) the validity of the estimated cointegrating equation by making sure that the left-hand side variable is participating in the cointegrating equation. 3) be able to critically evaluate research papers using this methodology.

ARCH(Auto-Regressive Conditional Heteroscedasticity) • An approach to modelling time-varying variance of a time series. (t2 : conditional variance) • Mostly financial market applications: the risk premium defined as a function of time-varying volatility (GARCH-in-mean); option pricing; leptokurtosis, volatility clustering. More efficient estimators can be obtained if heteroscedasticity in error terms is handled properly. ARCH: Engle (1982), GARCH: Bollerslev (1986), Taylor (1986).

ARCH(p) model: Mean Equation: yt = a + toryt = a + bXt + t ARCH(1): t2 =  + 2t-1 + t > 0,  >0 tis i.i.d. • GARCH(p,q) model: GARCH(2,1): t2 =  + 12t-1 + 22t-2 + 2t-1 +t  > 0, >0, >0 Exogenous or predetermined regressors can be added to the ARCH equations. The unconditional variance from a GARCH (1,1) model: 2 =  / [1-(+)] +< 1, otherwise nonstationary variance, which requires IGARCH.

Step 1. ARCH tests(H0: homoscedasticity) Heteroscedasticity tests: White test, Breusch-Pagan test (identifies changing variance due to regressors) ARCH-LM test: identifies only ARCH-type (auto-regressive conditional) heteroscedasticity • Step 2. Estimate a GARCH model (embedded in the mean equation) • Step 3. Using the estimated GARCH model, to forecast one-step ahead variance.

Asymmetric GARCH (TARCH or GJR Model) Leverage Effect: In stock markets, the volatility tends to increase when the market is falling, and decrease when it is rising. To model asymmetric effects on the volatility: t2 =  + 2t-1 + It-12t-1 + 2t-1 +t It-1 = { 1 if t-1 < 0, 0 if t-1 > 0 } If  issignificant, then we have asymmetric volatility effects. If  is significantly positive, it provides evidence for the leverage effect.

Multivariate GARCH If the variance ofa variable is affected by the past shocks to the variance of another variable, then a univariate GARCH specification suffers from an omitted variable bias. VECH Model: (describes the variance and covariance as a function of past squared error terms, cross-product error terms, past variances and past covariances.) 1,t2 = 1 + 1,121,t-1 + 1,222,t-1 + 1,31,t-12,t-1 + 1,121,t-1 + 1,222,t-1 +1,3Cov1,2,t-1 +1,t 2,t2 = 2 + 2,121,t-1 + 2,222,t-1 + 2,31,t-12,t-1 + 2,121,t-1 + 2,222,t-1 +2,3Cov1,2,t-1 +2,t Cov1,2,t = 3 + 3,121,t-1 + 3,222,t-1 + 3,31,t-12,t-1 + 3,121,t-1 + 3,222,t-1 +3,3Cov1,2,t-1 +3,t Two key terms: Shock spillover, Volatility spillover

Diagonal VECH Model: (describes the variance as a function of past squared error term and variance; and describes the covariance as a function of past cross-product error terms and past covariance.) 1,t2 = 1 + 1,121,t-1 + 1,121,t-1 + 1,t 2,t2 = 2 + 2,222,t-1 + 2,222,t-1 +2,t Cov1,2,t = 3 + 3,31,t-12,t-1 + 3,3Cov1,2,t-1 +3,t This one is less computationally-demanding, but still cannot guarantee positive semi-definite covariance matrix. BEKK Model: guarantees the positive definiteness.

ECO400 – PART I: TIME SERIES