Introduction to Economic Time Series

Transcription

1 Econometrics II Introduction to Economic Time Series Morten Nyboe Tabor

2 Learning Goals 1 Give an account for the important differences between (independent) cross-sectional data and time series data. 2 Give a precise definition and interpretation of the concept of stationarity of time series data, and explain the consequences of stationarity and weak dependence of a stochastic process. 3 Evaluate and justify if a time series is stationary based on a graphical analysis. Econometrics II Introduction to Economic Time Series Slide 2

3 Outline 1 The difference between cross-sectional and time series data 2 Stationarity 3 Measuring time dependence 4 Characteristics of economic time series data 5 Non-stationarity and transformations to stationarity Econometrics II Introduction to Economic Time Series Slide 3

4 1. The difference between cross-sectional and time series data

5 Random Sampling (Cross-Sectional Data) The observations x 1,..., x N are randomly drawn from a fixed population. N observations from the same distribution. No ordering. Random Sampling (Cross-Sectional Data) The observations 1 are randomly drawn from a fixed population. When Observations N is large we fromcan the same characterize distribution. the Noordering. distribution. Law of large numbers (for I.I.D. variables): N 1 N xi E(xi). i=1 When is large we can characterize the distribution. Law of large numbers: X 1 ( ). =1 Random sampling from population Sample distribution approximates the population y y y 2 y 1 y 3 y 8 y 7 y 4 y 9 y 6 y 5 5of17 Econometrics II Introduction to Economic Time Series Slide 5

6 Time Series Data A time series is a set of observations y 1, y 2,..., y t,..., y T, where t is the time index. Natural temporal ordering: 1 < 2 <... < t <... < T. It holds that y t 1 is realized (and often observed) when y t is determined. We often focus on conditional models y t y t 1, y t 2,... E.g. a model for the conditional mean, or for the entire conditional distribution. Most data in macro-economics and finance come in this form. Very different characteristics. Crucial: The tools should match the characteristics of the data! Econometrics II Introduction to Economic Time Series Slide 6

7 Stochastic Processes (Time Series Data) Stochastic Processes (Time Series Data) ervation is a realization of a random variable. Observation y t is a realization of a random variable y y one observation per random variable! t. Only one observation per random variable! The sequence of random variables y 1,..., y T is denoted a stochastic process. sequence of random variables 1 is denoted a stochastic process. y t Econometrics II Introduction to Economic Time Series Slide 7

8 If we could rerun history M times: Stochastic process: y 1, y 2,..., y t,..., y T Realization 1: y (1) 1, y (1) (1) 2,..., y t,..., y (1) T.... Realization m: y (m) 1, y (m) 2,..., y (m) t,..., y (m) T.... Realization M: y (M) 1, y (M) 2,..., y (M) t,..., y (M) T Define the ensemble mean, E (y t). Estimated with the cross-sectional average Ê(y t) = 1 M M m=1 y (m) t. Fundamentally different from the time average of a realized sample path y T = 1 T T y t. t=1 Econometrics II Introduction to Economic Time Series Slide 8

9 2. Stationarity

10 Stationarity Strict stationarity A time series, y 1, y 2,..., y t,..., y T, is called strictly stationary if the distributions of (y t, y t+1,..., y t+s) and (y t+h, y t+1+h,..., y t+s+h ) are the same for all h. The distribution of y t does not depend on t. Weak stationarity The time series is called weakly stationary if E(y t) = µ V (y t) = E((y t µ) 2 ) = γ 0 Cov(y t, y t h ) = E((y t µ) (y t h µ)) = γ h for h = 1, 2,... Econometrics II Introduction to Economic Time Series Slide 10

11 Stationarity y Strict stationarity implies that y t (t = 1, 2,..., T ) are random draws from the same unconditional distribution. Weak stationary is only a statement about the mean and the variance, i.e. the first two moments of the distribution. And y t fluctuates around a constnat level: equilibrium correcting / mean reverting. t Econometrics II Introduction to Economic Time Series Slide 11

12 Simulation Illustration We consider three simulated time series: We plot the simulated series along with the sample average. We increase the sample size by adding new observations and recursively update the sample average. What do you see? Econometrics II Introduction to Economic Time Series Slide 12

13 Simulation Illustration: What Do We See? (A) IID observations: y 1t = t y 1t Recursive sample average, ȳ 1 (T ) (B) Recursive sample average of y 1t (C) Stationary AR(1): y 2t =0.85 y 2t 1 t y 2t Recursive sample average, ȳ 2 (T ) 1 (D) Recursive sample average of y 2t (E) Non-stat. AR(1): y 3t = t (F) Recursive sample average of y 15 3t Recursive sample average, ȳ 3 (T ) y 3t Recursive sample average, ȳ 3 (T ) Page: 1 of 1 Econometrics II Introduction to Economic Time Series Slide 13

14 Why do we care if a time series process is stationary and weakly dependent? Econometrics II Introduction to Economic Time Series Slide 14

15 Stationarity: Main Result Strict stationarity implies that y t (t = 1, 2,..., T ) contains information about the same distribution. And, in terms of weak stationarity, y t fluctuates around a constant level: equilibrium correcting. To estimate parameters, we need a law of a large numbers (LLN) to hold. We make an additional technical assumption called weak dependence: Observations y t and y t h becomes approximately independent for h. Then y is a consistent estimator of E(y t). Given stationarity and weak dependence of y t and x t, most properties of OLS in the IID case carry over to the time series regression y t = x t β + ɛ t. We return to this issue later. Compared to the I.I.D. case, weak stationarity replaces the condition that observations are Identically Distributed, whereas weak dependence replaces the Independence condition. The most important distinction in time series econometrics is whether the time series of interest are stationary or not! Determines which methods we should use. Econometrics II Introduction to Economic Time Series Slide 15

16 3. Measuring time dependence

17 Measuring Time Dependence A characteristic feature of economic time series is a clear dependence over time. We can measure the dependence by the correlation Corr(y t, y t h ) = Cov(yt, y t h), h =..., 2, 1, 0, 1, 2,... V (yt) V (y t h ) As a function of h this is called a correlogram. Under stationarity the formula simplifies to ρ h = Cov(yt, y t h), h =..., 2, 1, 0, 1, 2,... V (y t) called the autocorrelation function (ACF). You may think of weak dependence as ρ h 0 as h. Both can be estimated by replacing population moments with sample moments: Ĉov(y t, y t h ) = 1 T h T (y t y) (y t h y). t=h+1 Under stationarity the associated estimators are asymptotically equivalent. Econometrics II Introduction to Economic Time Series Slide 17

18 Empirical Example Look at the US unemployment rate. Empirical Example Look at It the is US notunemployment clear whether urate. t equilibrium corrects. It is not clear whether equilibrium corrects. It fluctuates It fluctuates within within bounds, bounds, but but deviations deviations are very are very persistent persistent and and equilibrium correction is very slow. equilibrium correction is very slow. u t and u t h are highly correlated for large values of h. and are highly correlated for large values of. We return to formal testing later. We return to formal testing later (A) US unemployment rate 1.0 (B) ACF for (A) Econometrics II Introduction to Economic Time Series 13 Slide of 17 18

19 4. Characteristics of economic time series data

20 What arecharacteristics the characteristics of of economic Timetime Series seriesdata data? 12.5 (A) US unemployment rate 1.00 (B) Danish productivity (logs) (C) Danish income and consumption (logs) Income Consumption (D) Daily change in the NASDAQ index (%) Period: 3/ to 26/ of17 Econometrics II Introduction to Economic Time Series Slide 20

21 5. Non-stationarity and transformations to stationarity

22 Transformations to Stationarity Many economic time series are not stationary. But sometimes a non-stationary time series can be transformed to stationarity. Three important cases: (A) Remove a deterministic trend. Trend stationary. (B) Take first differences. Difference stationary or integrated or first order. (C) Combine several variables. Cointegration. Econometrics II Introduction to Economic Time Series Slide 22

23 (A) If the non-stationarity of is due to a deterministic trend, then the de-trended variable (A) If the non-stationarity of y t is due to a deterministic trend, then the de-trended variable = 0 1 might be stationary. In this case, yt = y t µ 0 µ is called trend-stationary. 1t, Themight de-trended be stationary. variable can In be thisfound case, asy t the is estimated called trend-stationary. residual in the linear regression The de-trended variable can be found as the estimated residual in the linear regression = As an example look at the Danish y t = productivity. µ 0 + µ 1t + yt. As an example look at the Danish productivity: (C) Danish productivity (log) minus trend 1.0 (D) ACF for (C) of 17 Econometrics II Introduction to Economic Time Series Slide 23

24 (B) Alternatively it might turn out that y t is non-stationary while the first (B) Alternatively difference, it might turn out that is non-stationary while the first difference, y t = y t y t 1, is stationary. In this case, y t is difference = stationary 1 or integrated of first is stationary. order, I(1). In this case, is difference stationary or integrated of first order, I(1). As an example look at Danish private consumption. As an example look at Danish private consumption: (E) change in Danish consumption (log) 1 (F) ACF for (E) Econometrics II Introduction to Economic Time Series Slide 24

25 (C) Finally, it might turn out that two variables, y t and x t, are non-stationary, (C) Finally, but it related mightso turn that outa that linear two combination variables, and, are non-stationary, but related so that a linear combination z t = y t β x t, = is stationary. Here y t and x t are I(1) but so-called co-integrated. is stationary. Here and are I(1) but so-called co-integrated. As an example consider consumption, c t, and income, y t. Both are I(1) As an and example have no consider equilibrium. consumption, They are related,,andincome, however,. Both and the are savings I(1) and rate, have no equilibrium. They are related, however, and the savings rate, s t = y t c t, = seems to be stationary and equilibrium corrects. seems to be stationary and equilibrium corrects. (G) Danish savings rate (log) 1 (H) ACF for (G) of 17 Econometrics II Introduction to Economic Time Series Slide 25

26 Learning Goals, Again 1 Give an account of the important differences between (independent) cross-sectional data and time series data. 2 Give a precise definition and interpretation of the concept of stationarity and weak dependence, and explain the important consequences of stationarity and weak dependence of a time series process. 3 Evaluate and justify if a time series is stationary based on a graphical analysis. Next: The linear regression model for stationary time series. Econometrics II Introduction to Economic Time Series Slide 26