Part 10: Time Series Applications [ 1/64] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

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Part 10: Time Series Applications [ 4/64] Dear professor Greene, I have a plan to run a (endogenous or exogenous) switching regression model for a panel data set. To my knowledge, there is no routine for this in other software, and I am not so good at coding a program. Fortunately, I am advised that LIMDEP has a built in function (or routine) for the panel switch model.

Part 10: Time Series Applications [ 5/64] Endogenous Switching (ca.1980) Not identified. Regimes do not coexist.

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Part 10: Time Series Applications [ 7/64]

Part 10: Time Series Applications [ 8/64] Modeling an Economic Time Series  Observed y 0, y 1, …, y t,…  What is the “sample”  Random sampling?  The “observation window”

Part 10: Time Series Applications [ 9/64] Estimators  Functions of sums of observations  Law of large numbers? Nonindependent observations What does “increasing sample size” mean?  Asymptotic properties? (There are no finite sample properties.)

Part 10: Time Series Applications [ 10/64] Interpreting a Time Series  Time domain: A “process” y(t) = ax(t) + by(t-1) + … Regression like approach/interpretation  Frequency domain: A sum of terms y(t) = Contribution of different frequencies to the observed series.  (“High frequency data and financial econometrics – “frequency” is used slightly differently here.)

Part 10: Time Series Applications [ 11/64] For example,…

Part 10: Time Series Applications [ 12/64] In parts…

Part 10: Time Series Applications [ 13/64] Studying the Frequency Domain  Cannot identify the number of terms  Cannot identify frequencies from the time series  Deconstructing the variance, autocovariances and autocorrelations Contributions at different frequencies Apparent large weights at different frequencies Using Fourier transforms of the data Does this provide “new” information about the series?

Part 10: Time Series Applications [ 14/64] Autocorrelation in Regression  Y t = b’x t + ε t  Cov(ε t, ε t-1 ) ≠ 0  Ex. RealCons t = a + bRealIncome + ε t U.S. Data, quarterly,

Part 10: Time Series Applications [ 15/64] Autocorrelation  How does it arise?  What does it mean?  Modeling approaches Classical – direct: corrective  Estimation that accounts for autocorrelation  Inference in the presence of autocorrelation Contemporary – structural  Model the source  Incorporate the time series aspect in the model

Part 10: Time Series Applications [ 16/64] Stationary Time Series  z t = b 1 y t-1 + b 2 y t-2 + … + b P y t-P + e t  Autocovariance: γ k = Cov[y t,y t-k ]  Autocorrelation:  k = γ k / γ 0  Stationary series: γ k depends only on k, not on t Weak stationarity: E[y t ] is not a function of t, E[y t * y t-s ] is not a function of t or s, only of |t-s| Strong stationarity: The joint distribution of [y t,y t-1,…,y t-s ] for any window of length s periods, is not a function of t or s.  A condition for weak stationarity: The smallest root of the characteristic polynomial: 1 - b 1 z 1 - b 2 z 2 - … - b P z P = 0, is greater than one. The unit circle Complex roots Example: y t =  y t-1 + e e, 1 -  z = 0 has root z = 1/ , | z | > 1 => |  | < 1.

Part 10: Time Series Applications [ 17/64] Stationary vs. Nonstationary Series

Part 10: Time Series Applications [ 18/64] The Lag Operator  Lc = c when c is a constant  Lx t = x t-1  L 2 x t = x t-2  L P x t + L Q x t = x t-P + x t-Q  Polynomials in L: y t = B(L)y t + e t  A(L) y t = e t  Invertibility: y t = [A(L)] -1 e t

Part 10: Time Series Applications [ 19/64] Inverting a Stationary Series  y t =  y t-1 + e t  (1-  L)y t = e t  y t = [1-  L] -1 e t = e t +  e t-1 +  2 e t-2 + … Stationary series can be inverted Autoregressive vs. moving average form of series

Part 10: Time Series Applications [ 20/64] Regression with Autocorrelation  y t = x t ’b + e t, e t =  e t-1 + u t  (1-  L)e t = u t  e t = (1-  L) -1 u t E[e t ] = E[ (1-  L) -1 u t ] = (1-  L) -1 E[u t ] = 0 Var[e t ] = (1-  L) -2 Var[u t ] = 1+  2  u 2 + … =  u 2 /(1-  2 ) Cov[e t,e t-1 ] = Cov[  e t-1 + u t, e t-1 ] = =  Cov[e t-1,e t-1 ]+Cov[u t,e t-1 ] =   u 2 /(1-  2 )

Part 10: Time Series Applications [ 21/64] OLS vs. GLS  OLS Unbiased? Consistent: (Except in the presence of a lagged dependent variable) Inefficient  GLS Consistent and efficient

Part 10: Time Series Applications [ 22/64] | Ordinary least squares regression | | LHS=REALCONS Mean = | | Autocorrel Durbin-Watson Stat. = | | Rho = cor[e,e(-1)] = | |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| Constant REALDPI | Robust VC Newey-West, Periods = 10 | Constant REALDPI | AR(1) Model: e(t) = rho * e(t-1) + u(t) | | Final value of Rho = | | Iter= 6, SS= , Log-L= | | Durbin-Watson: e(t) = | | Std. Deviation: e(t) = | | Std. Deviation: u(t) = | | Durbin-Watson: u(t) = | | Autocorrelation: u(t) = | | N[0,1] used for significance levels | |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Constant REALDPI RHO

Part 10: Time Series Applications [ 23/64] Detecting Autocorrelation  Use residuals Durbin-Watson d= Assumes normally distributed disturbances strictly exogenous regressors  Variable addition (Godfrey) y t =  ’x t +  ε t-1 + u t Use regression residuals e t and test  = 0 Assumes consistency of b.

Part 10: Time Series Applications [ 24/64] A Unit Root?  How to test for  = 1?  By construction: ε t – ε t-1 = (  - 1)ε t-1 + u t Test for γ = (  - 1) = 0 using regression? Variance goes to 0 faster than 1/T. Need a new table; can’t use standard t tables. Dickey – Fuller tests  Unit roots in economic data. (Are there?) Nonstationary series Implications for conventional analysis

Part 10: Time Series Applications [ 25/64] Reinterpreting Autocorrelation

Part 10: Time Series Applications [ 26/64] Integrated Processes  Integration of order (P) when the P’th differenced series is stationary  Stationary series are I(0)  Trending series are often I(1). Then y t – y t-1 =  y t is I(0). [Most macroeconomic data series.]  Accelerating series might be I(2). Then (y t – y t-1 )- (y t – y t-1 ) =  2 y t is I(0) [Money stock in hyperinflationary economies. Difficult to find many applications in economics]

Part 10: Time Series Applications [ 27/64] Cointegration: Real DPI and Real Consumption

Part 10: Time Series Applications [ 28/64] Cointegration – Divergent Series?

Part 10: Time Series Applications [ 29/64] Cointegration  X(t) and y(t) are obviously I(1)  Looks like any linear combination of x(t) and y(t) will also be I(1)  Does a model y(t) = bx(t) + u(u) where u(t) is I(0) make any sense? How can u(t) be I(0)?  In fact, there is a linear combination, [1,-  ] that is I(0).  y(t) =.1*t + noise, x(t) =.2*t + noise  y(t) and x(t) have a common trend  y(t) and x(t) are cointegrated.

Part 10: Time Series Applications [ 30/64] Cointegration and I(0) Residuals

Part 10: Time Series Applications [ 31/64] Cross Country Growth Convergence

Part 10: Time Series Applications [ 32/64] Heterogeneous Dynamic Model

Part 10: Time Series Applications [ 33/64] “Fixed Effects” Approach

Part 10: Time Series Applications [ 34/64] Country Means

Part 10: Time Series Applications [ 35/64] Country Means (cont.)

Part 10: Time Series Applications [ 36/64] Time Series

Part 10: Time Series Applications [ 37/64] Pooling Essentially the same as the time series case. OLS or GLS are inconsistent There could be no instrument that would work (by construction)

Part 10: Time Series Applications [ 38/64] A Mixed/Fixed Approach

Part 10: Time Series Applications [ 39/64] A Mixed Fixed Model Estimator

Part 10: Time Series Applications [ 40/64] Nair-Reichert and Weinhold on Growth Weinhold (1996) and Nair–Reichert and Weinhold (2001) analyzed growth and development in a panel of 24 developing countries observed for 25 years, 1971–1995. The model they employed was a variant of the mixed-fixed model proposed by Hsiao (1986, 2003). In their specification, GGDP i,t = α i + γ i d it GGDP i,t-1 + β 1i GGDI i,t-1 + β 2i GFDI i,t-1 + β 3i GEXP i,t-1 + β 4 INFL i,t-1 + ε i,t GGDP = Growth rate of gross domestic product, GGDI = Growth rate of gross domestic investment, GFDI = Growth rate of foreign direct investment (inflows), GEXP = Growth rate of exports of goods and services, INFL = Inflation rate. The constant terms and coefficients on the lagged dependent variable are country specific. The remaining coefficients are treated as random, normally distributed, with means β k and unrestricted variances. They are modeled as uncorrelated. The model was estimated using a modification of the Hildreth–Houck–Swamy method

Part 10: Time Series Applications [ 41/64] Analysis of Macroeconomic Data  Integrated series  The problem with regressions involving nonstationary series Spurious regressions Unit roots and misleading relationships  Solutions to the “problem” Random walks and first differencing Removing common trends  Cointegration: Formal solutions to regression models involving nonstationary data  Extending these results to panels Large T and small T cases. Parameter heterogeneity across countries

Part 10: Time Series Applications [ 42/64] Nonstationary Data

Part 10: Time Series Applications [ 43/64] Integrated Series

Part 10: Time Series Applications [ 44/64] Stationary Data

Part 10: Time Series Applications [ 45/64] Unit Root Tests

Part 10: Time Series Applications [ 46/64] KPSS Test-1

Part 10: Time Series Applications [ 47/64] KPSS Test-2

Part 10: Time Series Applications [ 48/64] Cointegrated Variables?

Part 10: Time Series Applications [ 49/64] Cointegrating Relationships  Implications: Long run vs. short run relationships Problems of spurious regressions (as usual)  Problem for existing empirical studies: Regressions involving variables of different integration. E.g., regressions of flows on stocks

Part 10: Time Series Applications [ 50/64] Money demand example

Part 10: Time Series Applications [ 51/64] Panel Unit Root Tests

Part 10: Time Series Applications [ 52/64] Implications  Separate analyses by country  How to combine data and test statistics  Cointegrating relationships across countries

Part 10: Time Series Applications [ 53/64] Purchasing Power Parity

Part 10: Time Series Applications [ 54/64] Application “Some international evidence on price determination: a non-stationary panel Approach,” Paul Ashworth, Joseph P. Byrne, Economic Modelling, 20, 2003, p quarters, 13 OECD countries log p i,t = β 0 + β 1 log(unit labor cost i,t ) + β 2 log(world price,t ) + β 3 log(intermediate goods price i,t ) + β 4 (log-output gap i,t ) + ε i,t Various tests for unit roots and cointegration

Part 10: Time Series Applications [ 55/64] Vector Autoregression The vector autoregression (VAR) model is one of the most successful, flexible, and easy to use models for the analysis of multivariate time series. It is a natural extension of the univariate autoregressive model to dynamic multivariate time series. The VAR model has proven to be especially useful for describing the dynamic behavior of economic and financial time series and for forecasting. It often provides superior forecasts to those from univariate time series models and elaborate theory-based simultaneous equations models. Forecasts from VAR models are quite flexible because they can be made conditional on the potential future paths of specified variables in the model. In addition to data description and forecasting, the VAR model is also used for structural inference and policy analysis. In structural analysis, certain assumptions about the causal structure of the data under investigation are imposed, and the resulting causal impacts of unexpected shocks or innovations to specified variables on the variables in the model are summarized. These causal impacts are usually summarized with impulse response functions and forecast error variance decompositions. Eric Zivot:

Part 10: Time Series Applications [ 56/64] VAR

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Part 10: Time Series Applications [ 59/64] Zivot’s Data

Part 10: Time Series Applications [ 60/64] Impulse Responses

Part 10: Time Series Applications [ 61/64] GARCH Models: A Model for Time Series with Latent Heteroscedasticity Bollerslev/Ghysel, 1974

Part 10: Time Series Applications [ 62/64] ARCH Model

Part 10: Time Series Applications [ 63/64] GARCH Model

Part 10: Time Series Applications [ 64/64] Estimated GARCH Model GARCH MODEL Dependent variable Y Log likelihood function Restricted log likelihood Chi squared [ 2 d.f.] Significance level McFadden Pseudo R-squared Estimation based on N = 1974, K = 4 GARCH Model, P = 1, Q = 1 Wald statistic for GARCH = Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X |Regression parameters Constant| |Unconditional Variance Alpha(0)|.01076*** |Lagged Variance Terms Delta(1)|.80597*** |Lagged Squared Disturbance Terms Alpha(1)|.15313*** |Equilibrium variance, a0/[1-D(1)-A(1)] EquilVar|