Introduction to Time Series Regression and Forecasting (SW Chapter 12)

Transcription

1 Introduction to Time Series Regression and Forecasting (SW Chapter 12) Time series data are data collected on the same observational unit at multiple time periods Aggregate consumption and GDP for a country (for example, 20 years of quarterly observations = 80 observations) Yen/$, pound/$ and Euro/$ exchange rates (daily data for 1 year = 365 observations) Cigarette consumption per capita for a state 12-1

2 Example #1 of time series data: US rate of inflation 12-2

3 Example #2: US rate of unemployment 12-3

4 Why use time series data? To develop forecasting models o What will the rate of inflation be next year? To estimate dynamic causal effects o If the Fed increases the Federal Funds rate now, what will be the effect on the rates of inflation and unemployment in 3 months? in 12 months? o What is the effect over time on cigarette consumption of a hike in the cigarette tax Plus, sometimes you don t have any choice o Rates of inflation and unemployment in the US can be observed only over time. 12-4

5 Time series data raises new technical issues Time lags Correlation over time (serial correlation or autocorrelation) Forecasting models that have no causal interpretation (specialized tools for forecasting): o autoregressive (AR) models o autoregressive distributed lag (ADL) models Conditions under which dynamic effects can be estimated, and how to estimate them Calculation of standard errors when the errors are serially correlated 12-5

6 Using Regression Models for Forecasting (SW Section 12.1) Forecasting and estimation of causal effects are quite different objectives. For forecasting, o 2 R matters (a lot!) o Omitted variable bias isn t a major problem! o We will not worry about interpreting coefficients in forecasting models o External validity is paramount: the model estimated using historical data must hold into the (near) future 12-6

7 Introduction to Time Series Data and Serial Correlation (SW Section 12.2) First we must introduce some notation and terminology. Notation for time series data Y t = value of Y in period t. Data set: Y 1,,Y T = T observations on the time series random variable Y We consider only consecutive, evenly-spaced observations (for example, monthly, 1960 to 1999, no missing months) (else yet more complications...) 12-7

8 We often transform time series variables using lags, first differences, logarithms, & growth rates 12-8

9 Example: Quarterly rate of inflation at an annual rate CPI in the first quarter of 1999 (1999:I) = CPI in the second quarter of 1999 (1999:II) = Percentage change in CPI, 1999:I to 1999:II = = = 0.703% Percentage change in CPI, 1999:I to 1999:II, at an annual rate = 4x0.703 = 2.81% (percent per year) Like interest rates, inflation rates are (as a matter of convention) reported at an annual rate. Using the logarithmic approximation to percent changes yields 4x100x[log(166.03) log(164.87)] = 2.80% 12-9

10 Example: US CPI inflation its first lag and its change CPI = Consumer price index (Bureau of Labor Statistics) 12-10

11 Autocorrelation The correlation of a series with its own lagged values is called autocorrelation or serial correlation. The first autocovariance of Y t is cov(y t,y t 1 ) The first autocorrelation of Y t is corr(y t,y t 1 ) These are related by cov( Y, Y ) t t 1 corr(y t,y t 1 ) = var( Y ) var( Y ) t t 1 =ρ

12 These are population correlations they describe the population joint distribution of (Y t,y t j ) We are often interested in estimating these population parameters. The natural estimator: Sample autocorrelations 12-12

13 Sample autocorrelations The j th sample autocorrelation is an estimate of the j th population autocorrelation: ˆ ρ j = where covˆ( Y t, Y co vˆ( Y, Y t t T t j T 1 j 1 j ) t= j+ 1 ) / = varˆ( Y t ) ( Y Y )( Y Y t j+ 1, T t j 1, T j ) where Y j + 1, T is the sample average of Y t computed over observations t = j+1,,t o Note: the summation is over t=j+1 to T (why)? 12-13

14 Example: Autocorrelations of: (1) the quarterly rate of U.S. inflation (2) the quarter-to-quarter change in the quarterly rate of inflation 12-14

15 The inflation rate is highly serially correlated (ρ 1 =.85) Last quarter s inflation rate contains much information about this quarter s inflation rate The plot is dominated by multiyear swings But there are still surprise movements! 12-15

16 More examples of time series & transformations 12-16

17 Stationarity: a key idea for external validity of time series regression Stationarity says that the past is like the present and the future, at least in a probabilistic sense. We ll focus on the case that Y t stationary