Introduction to Time Series Heino Bohn Nielsen 1 of 15 Outline - PDF document

Econometrics 2 — Fall 2005 Introduction to Time Series Heino Bohn Nielsen 1 of 15 Outline (1) What is a time series? (2) Important characteristics of time series data. (3) Data sampling and stochastic processes. Cross-sectional data vs. time series data. (4) Stationarity. (5) Measuring time dependence. (6) Non-stationarity and transformations to stationarity. 2 of 15

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. • It holds that y t − 1 is observed when y t is determined. Focus on conditional models y t | y t − 1 , y t − 2 , ... . • Most data in macro-economics and fi nance come in this form. • Very di ff erent characteristics. The tools should match the characteristics of the data. 3 of 15 Characteristics of Time Series Data (A) US unemployment rate (B) Danish productivity (logs) 1.00 12.5 10.0 0.75 7.5 0.50 5.0 2.5 0.25 0.0 1950 1960 1970 1980 1990 2000 1970 1980 1990 2000 (C) Danish income and consumption (logs) (D) Daily change in the NASDAQ index (%) Income 6.4 Consumption 10 6.3 5 6.2 0 6.1 -5 6.0 5.9 -10 Period: 3/1-2000 to 26/2-2004 1970 1980 1990 2000 0 150 300 450 600 750 900 1050 4 of 15

Random Sampling (Cross-Sectional Data) • The observations x 1 , ..., x N are randomly drawn from a fi xed population. N Observations from the same distribution. No ordering. • When N is large we can characterize the distribution. Law of large numbers: N − 1 X N i =1 x i → E [ x i ] . Random sampling from population Sample distribution approximates the population y y y 1 y 8 y 7 y 4 y 9 y 6 y 5 y 2 y 3 5 of 15 Stochastic Processes (Time Series Data) • Observation y t is a realization of a random variable y t . Only one observation per random variable! • The sequence of random variables y 1 , ..., y T is denoted a stochastic process. y t 1 2 3 4 5 6 7 6 of 15

• If we could rerun history M times: Stochastic process: y 1 , y 2 , ..., y t , ..., y T y (1) y (1) y (1) y (1) Realization 1 : 1 , 2 , ..., t , ..., T . . . . . . . . . . . . y ( m ) y ( m ) y ( m ) y ( m ) Realization m : , , ..., , ..., 1 2 t T . . . . . . . . . . . . y ( M ) y ( M ) y ( M ) y ( M ) Realization M : , , ..., , ..., . 1 2 t T • Consider the ensemble mean, E [ y t ] , estimated with the average M X E [ y t ] = 1 b y ( m ) . t M m =1 • Fundamentally di ff erent from the time average of a realized sample path X T y = 1 y (1) t . T t =1 7 of 15 Stationarity • A time series, y 1 , y 2 , ..., y t , ..., y T , is called strictly stationary if the distributions ( y t 1 , y t 2 , ..., y t n ) ( y t 1 + h , y t 2 + h , ..., y t n + h ) and are the same for all h . The distribution of y t does not depend on t . • 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 − k ] = E [( y t − µ ) ( y t − k − µ )] = γ k for k = 1 , 2 , ... y t 1 2 3 4 5 6 7 8 of 15

Main Result Stationarity implies that y t ( t = 1 , 2 , ..., T ) contain information on the same distribution. And y t fl uctuates around a constant level: equilibrium correcting. • We make the additional assumption that y t and y t − k becomes approximately inde- pendent for k → ∞ . This technical assumption called weak dependence, ensures that a law of a large numbers (LLN) hold. 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 0 t β + � t . We return to this issue later. The most important distinction in time series econometrics is whether the time series of interest are stationary or not. 9 of 15 Measuring Time Dependence A characteristic feature of time series is a clear dependence over time. We can measure the dependence by the correlation Cov ( y t , y t − k ) p Corr ( y t , y t − k ) = , k = ..., − 2 , − 1 , 0 , 1 , 2 , ... V ( y t ) · V ( y t − k ) Under stationarity the formula simpli fi es to the so-called autocorrelation function (ACF) ρ k = Cov ( y t , y t − k ) , k = ..., − 2 , − 1 , 0 , 1 , 2 , ... V ( y t ) which can be estimated by e.g. X T 1 t = k +1 ( y t − y ) ( y t − k − y ) T − k b ρ k = . X T t = k +1 ( y t − k − y ) 2 1 T − k 10 of 15

Empirical Example Look at the US unemployment rate. • It is not clear whether u t equilibrium corrects. • It fl uctuates within bounds, but deviations are very persistent and equilibrium cor- rection is very slow. u t and u t − k are highly correlated for large values if k . • We return to formal testing later. (A) US unemployment rate (B) ACF for (A) 1.0 10.0 7.5 0.5 5.0 1950 1960 1970 1980 1990 2000 0 5 10 15 11 of 15 Transformation to Stationarity Many economic time series are not stationary. Sometimes, a non-stationary time series can be transformed to stationarity. Three important cases: (A) Remove a deterministic trend. Trend stationary. (B) Take fi rst di ff erences. Di ff erence stationary or integrated or fi rst order. (C) Combine several variables. Cointegration. 12 of 15

(A) If the non-stationary of y t is due to a deterministic trend, then the de-trended variable y ∗ t = y t − µ 0 − µ 1 t, might be stationary. In this case, y t is called trend-stationary. The de-trended variable can be found as the estimated residual in the linear regression y t = µ 0 + µ 1 t + y ∗ t . As an example look at the Danish productivity. (C) Danish productivity (log) minus trend (D) ACF for (C) 1.0 0.05 0.00 0.5 -0.05 1970 1980 1990 2000 0 5 10 15 13 of 15 (B) Alternatively it might turn out that y t is non-stationary while the fi rst di ff erence, ∆ y t = y t − y t − 1 , is stationary. In this case, y t is di ff erence stationary or integrated of fi rst order, I(1). As an example look at Danish private consumption. (E) change in Danish consumption (log) (F) ACF for (E) 1 0.05 0 0.00 -0.05 1970 1980 1990 2000 0 5 10 15 14 of 15

(C) Finally, it might turn out that two variables, y t and x t , 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. As an example consider consumption, c t , and income, y t . Both are I(1) and have no equilibrium. They are related, however, and the savings rate, s t = y t − c t , seems to be stationary and equilibrium corrects. (G) Danish savings rate (log) (H) ACF for (G) 1 0.15 0.10 0 0.05 0.00 1970 1980 1990 2000 0 5 10 15 15 of 15