Economics 300 Econometrics Time Series
advertisement
Review Serial Correlation Time Series Economics 300 Econometrics Time Series Dennis C. Plott University of Illinois at Chicago Department of Economics www.dennisplott.com Fall 2014 Dennis C. Plott (UIC) ECON 300 – Fall 2014 1 / 29 Review Serial Correlation OLS (Gauss-Markov) Assumptions Time Series Ï One of the assumptions of the error term (u) in a regression analysis is that the error must be independent and identically distributed (i.i.d.). Ï Error variance is the same for all values; i.e., homoskedasticity. Ï Ï Ï Error is not related to other error values; i.e., no serial correlation. Ï Ï Ï Ï covered in this section similar consequences as heteroskedasticity can use similar solution, but is typically dealt with by incorporating into model Error is normally distributed. Ï Dennis C. Plott (UIC) violation consequences: biased standard errors (invalid inference); i.e., t- and F-statistics useless solutions: simple and reliable fix – use robust standard errors or cluster the standard errors, then inference is valid optional ECON 300 – Fall 2014 2 / 29 Review Serial Correlation Serial Correlation – Introduction Time Series Ï Ï If a systematic correlation does exist between one observation of the error term and another, then it will be more difficult for OLS to get accurate estimates of the standard errors of the coefficients This assumption is most likely to be violated in time-series models: Ï Ï Ï An increase in the error term in one time period (a random shock, for example) is likely to be followed by an increase in the next period. Example: Hurricane Katrina If, over all the observations of the sample ut+1 (i.e., next period’s erorr) is correlated with ut (i.e., this period’s error), then the error term is said to be serially correlated (or autocorrelated), and the OLS (Gauss-Markov) assumptions are violated. Dennis C. Plott (UIC) ECON 300 – Fall 2014 3 / 29 Review Serial Correlation Time Series (Pure) Serial Correlation Ï Ï Serial correlation (pure) occurs when the OLS (Gauss-Markov) assumption of uncorrelated observations of the error term is violated (in a correctly specified equation). The most commonly assumed kind of serial correlation is first-order serial correlation, in which the current value of the error term is a function of the previous value of the error term: ut = ρut−1 + vt where Ï Ï Ï Ï u = the error term of the equation in question ρ = the first-order autocorrelation coefficient v = an i.i.d. (not serially correlated) error term; i.e., OLS (Gauss-Markov) assumptions not violated (Impure) Serial Correlation is serial correlation that is caused by a specification error such as: Ï Ï omitted variable(s) and/or incorrect functional form Dennis C. Plott (UIC) ECON 300 – Fall 2014 4 / 29 Review Serial Correlation Time Series (Pure) Serial Correlation (Continued) Ï The magnitude of ρ indicates the strength of the serial correlation: Ï Ï Ï Ï If ρ is zero, there is no serial correlation As ρ approaches one in absolute value, the previous observation of the error term becomes more important in determining the current value of ut and a high degree of serial correlation exists For ρ to exceed one is unreasonable (i.e., unlikely), since the error term effectively would “explode” As a result of this, we can state, in general, that: −1 < ρ < +1 Ï The sign of ρ indicates the nature of the serial correlation in an equation Ï Positive Ï Ï Negative Ï Dennis C. Plott (UIC) implies that the error term tends to have the same sign from one time period to the next implies that the error term has a tendency to switch signs from negative to positive and back again in consecutive observations ECON 300 – Fall 2014 5 / 29 Review Serial Correlation Serial Correlation – Graphs Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 6 / 29 Review Serial Correlation (Pure) Serial Correlation – Consequences Time Series Ï The existence of serial correlation in the error term of an equation violates the Gauss-Markov assumptions and the estimation of the equation with OLS has at least three consequences: 1. Pure serial correlation does NOT cause bias in the coefficient estimates. 2. Serial correlation causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators); i.e., OLS in no longer BLUE. 3. Serial correlation causes the OLS estimates of the standard error to be biased, leading to unreliable hypothesis testing. Ï Dennis C. Plott (UIC) Typically the bias in the SE estimate is negative (downward), meaning that OLS underestimates the standard errors of the coefficients (and thus overestimates the t-scores) ECON 300 – Fall 2014 7 / 29 Review Serial Correlation (Pure) Serial Correlation – One Solution: Newey-West SE Time Series Ï Ï Newey-West standard errors take account of serial correlation by correcting the standard errors without changing the estimated coefficients. The logic begin Newey-West standard errors: Ï If serial correlation does not cause bias in the estimated coefficients but does impact the standard errors, then it makes sense to adjust the estimated equation in a way that changes the standard errors but not the coefficients. Ï The Newey-West standard errors are biased but generally more accurate than uncorrected standard errors for large samples in the face of serial correlation. Ï As a result, Newey-West standard errors can be used for t-tests and other hypothesis tests in most samples without the errors of inference potentially caused by serial correlation. Ï Typically, Newey-West standard errors are larger than OLS standard errors, thus producing lower t-scores Dennis C. Plott (UIC) ECON 300 – Fall 2014 8 / 29 Review Serial Correlation Time Series Models Time Series Ï Time Series Examples Ï Ï Ï Ï Modeling relationships using data collected over time – prices, quantities, GDP, etc. Forecasting – predicting economic growth. Time series involves decomposition into a trend, seasonal, cyclical, and irregular component. Problems ignoring lags Ï Values of yt are affected by the values of y in the past. Ï Ï Dennis C. Plott (UIC) For example, the amount of money in your bank account in one month is related to the amount in your account in a previous month. Regression without lags fails to account for the relationships through time and overestimates the relationship between the dependent and independent variables. ECON 300 – Fall 2014 9 / 29 Review Serial Correlation Time Series White Noise Ï White noise describes the assumption that each element in a series is a random draw from a population with zero mean and constant variance. Ï Autoregressive (AR) and moving average (MA) models correct for violations of this white noise assumption. Dennis C. Plott (UIC) ECON 300 – Fall 2014 10 / 29 Review Serial Correlation Time Series Autoregressive (AR) Models Ï Autoregressive (AR) models are models in which the value of a variable in one period is related to its values in previous periods. Ï AR(p) is an autoregressive model with p lags: yt = µ + p X γi yt−i + ut i=1 where µ is a constant and γp is the coefficient for the lagged variable at time t − p. Ï AR(1) is expressed as: yt = µ + γ1 yt−1 + ut = µ + γ(Lyt ) + ut (1 − γL)yt = µ + ut Dennis C. Plott (UIC) ECON 300 – Fall 2014 11 / 29 Review Serial Correlation Autoregressive (AR) Models (Continued) Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 12 / 29 Review Serial Correlation Moving Average (MA) Models Time Series Ï Moving average (MA) models account for the possibility of a relationship between a variable and the residuals from previous periods. Ï MA(q) is a moving average model with q lags: y t = µ + εt + q X θi εt−i i=1 where θq is the coefficient for the lagged error term in time t − q. Ï MA(1) model is expressed as: yt = µ + εt + θ1 εt−1 Ï Note: some programs, like SAS (unlike Stata), model θ with a reverse sign Dennis C. Plott (UIC) ECON 300 – Fall 2014 13 / 29 Review Serial Correlation Moving Average (MA) Models (Continued) Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 14 / 29 Review Serial Correlation Autoregressive Moving Average (ARMA) Models Time Series Ï Autoregressive moving average (ARMA) models combine both p autoregressive terms and q moving average terms, also called ARMA(p, q). yt = µ + p X i=1 Dennis C. Plott (UIC) γi yt−i + ut + q X θi ut−i i=1 ECON 300 – Fall 2014 15 / 29 Review Serial Correlation Time Series Stationarity Ï Ï Ï Modeling an ARMA(p, q) process requires stationarity. A stationary process has a mean and variance that do not change over time and the process does not have trends. An AR(1) disturbance process: ut = ρut−1 + vt Ï is stationary if |ρ| < 1 and vt is white noise. Example of a time-series variable, is it stationary? Dennis C. Plott (UIC) ECON 300 – Fall 2014 16 / 29 Review Serial Correlation Time Series Differencing Ï Ï Ï When a variable yt is not stationary, a common solution is to use differenced variable: ∆yt = yt − yt−1 , for first-order differences The variable yt is integrated of order one, denoted I(1), if taking a first difference produces a stationary process. ARIMA(p, d, q) denotes an ARMA model with p autoregressive lags, q moving average lags, and a is differenced of order d. Dennis C. Plott (UIC) ECON 300 – Fall 2014 17 / 29 Review Serial Correlation Seasonality Time Series Ï Seasonality is a particular type of autocorrelation pattern where patterns occur every “season”, like monthly, quarterly, etc. Ï For example, quarterly data may have the same pattern in the same quarter from one year to the next. Ï Seasonality must also be corrected before a time series model can be fitted. Dennis C. Plott (UIC) ECON 300 – Fall 2014 18 / 29 Review Serial Correlation Time Series Dickey-Fuller Test for Stationarity Ï Dickey-Fuller Test Ï Assume an AR(1) model. The model is non-stationary or a unit root is present if |ρ| = 1. yt = ρyt−1 + ut yt − yt−1 = ρyt−1 − yt−1 + ut ∆yt = (ρ − 1)yt−1 + ut ∆yt = γyt−1 + ut Ï We can estimate the above model and test for the significance of the γ coefficient. Ï Ï Ï Dennis C. Plott (UIC) If the null hypothesis is not rejected, γ = 0, then γ is NOT stationary. Difference the variable and repeat the Dickey-Fuller test to see if the differenced variable is stationary. If the null hypothesis is rejected, γ > 0, then yt is stationary. Use the variable. Note: non-significance is mean stationarity. ECON 300 – Fall 2014 19 / 29 Review Serial Correlation Time Series Dickey-Fuller Test for Stationarity (Continued) Ï Augmented Dickey-Fuller test Ï In addition to the model above, a drift µ and additional lags of the dependent variable can be added. ∆yt = µ + γyt−1 + p−1 X φj ∆yt−j + ut j=1 Ï Ï The augmented Dickey-Fuller test evaluates the null hypothesis that γ = 0. The model will be non-stationary if γ = 0. Dickey-Fuller test with a time trend Ï The model with a time trend: ∆yt = µ + βt + γyt−1 + p−1 X φj ∆yt−j + ut j=1 Ï Dennis C. Plott (UIC) Test the null hypothesis that β = 0 and γ = 0. Again, the will be non-stationary or will have a unit root present if γ = 0. ECON 300 – Fall 2014 20 / 29 Review Serial Correlation Time Series Autocorrelation Function (ACF) and Partial Autocorrelation Function (ACF) Ï Autocorrelation function (ACF) Ï ACF is the proportion of the autocovariance of yt and yt−k to the variance of a dependent variable yt ACF(k) = ρ k = Ï Ï Ï Cov(yt , yt−k ) Var(yt ) The autocorrelation function ACF(k) gives the gross (total) correlation between yt and yt−k . For an AR(1) model, the ACF is ACF(k) = ρ k = γk . We state the function tails off. Partial autocorrelation function (PACF) Ï Ï Dennis C. Plott (UIC) PACF is the simple correlation between yt and yt−k minus the part explained by the intervening lags. For an AR(1) model, the PACF is γ for the first lag and then cuts off. ECON 300 – Fall 2014 21 / 29 Review Serial Correlation Time Series Autocorrelation Function (ACF) and Partial Autocorrelation Function (ACF) (Continued) ACF PACF Dennis C. Plott (UIC) AR(p) MA(q) ARMA(p, q) tails off cuts off at lag p cuts off at lag q tails off tails off tails off ECON 300 – Fall 2014 22 / 29 Review Serial Correlation Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 23 / 29 Review Serial Correlation Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 24 / 29 Review Serial Correlation Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 25 / 29 Review Serial Correlation Time Series Dennis C. Plott (UIC) ECON 300 – Fall 2014 26 / 29 Review Serial Correlation Diagnostics for ARIMA Models Time Series Ï Testing for White Noise Ï The Ljung-Box statistic: Q = T (T + 2) p X ρ 2k k=1 Ï Ï T −k where ρ k is the sample autocorrelation at lag k. Under the null hypothesis that the series is white noise (data are independently distributed), Q has a limiting χ2 distribution with p degrees-of-freedom. Goodness-of-Fit Ï Ï Dennis C. Plott (UIC) Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) are two measures goodness of fit. They measure the trade-off between model fit and complexity of the model. A lower AIC or BIC value indicates a better fit (more parsimonious model). ECON 300 – Fall 2014 27 / 29 Review Serial Correlation Time Series The Box-Jenkins Methodology for ARIMA Model Selection: Identification Ï Examine the time plot of the series. Ï Ï Ï Identify outliers, missing values, and structural breaks in the data. Non-stationary variables may have a pronounced trend or have changing variance. Transform the data if needed. Use logs, differencing, or detrending. Ï Ï Ï Examine the autocorrelation function (ACF) and partial autocorrelation function (PACF). Ï Ï Ï Dennis C. Plott (UIC) Using logs works if the variability of data increases over time. Differencing the data can remove trends. But over-differencing may introduce dependence when none exists. Compare the sample ACF and PACF to those of various theoretical ARMA models. Use properties of ACF and PACF as a guide to estimate plausible models and select appropriate p, d, and q. With empirical data, several models may need to be estimated. Differencing may be needed if there is a slow decay in the ACF. ECON 300 – Fall 2014 28 / 29 Review Serial Correlation Time Series The Box-Jenkins Methodology for ARIMA Model Selection: Estimation & Diagnostics Ï Estimation step Ï Ï Ï Estimate ARIMA models and examine the various coefficients. The goal is to select a stationary and parsimonious model that has significant coefficients and a good fit. Diagnostic checking step Ï If the model fits well, then the residuals from the model should resemble a while noise process. Ï Ï Ï Ï Dennis C. Plott (UIC) Check for normality looking at a histogram of the residuals Check for independence by examining the ACF and PACF of the residuals, which should look like a white noise. The Ljung-Box-Pierce statistic performs a test of the magnitude of the autocorrelations of the correlations as a group. Examine goodness of fit using the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC). Use most parsimonious model with lowest AIC and/or BIC. ECON 300 – Fall 2014 29 / 29
