计量经济学课件

WEEK 10: MACROECONOMETRICS Introduction

1.The concept of stationarity

2.Spurious regressions

3.Testing for unit roots

4.Cointegration analysis

1. S TATIONARITY

Conditions for t y to be a stationary time series process i. t E y constant t ii. t Var y constant t

iii. ,t t k Cov y y constant t and all k≠0 Autoregressive time series

1t t t y y

- Notice no constant and t is a white noise error term.

- AR(1) model – time series behaviour of t y is largely explained by its value in the previous period.

- Necessary condition for stationarity 1 , if , 1 series is explosive and if 1 have a unit root.

Example 1 – Stationary AR(1) Model

STATA code

set obs 500 /*set number of observations*/

gen time=_n /*create time trend*/

gen y=0 if time==1 /* first observation set y=0*/

gen e=rnormal(0, 1) /*create a random number*/

replace y=(0.67*y[_n-1])+e if time~=1 /*AR(1) model =0.67*/ twoway (line y time) /*line plot*/

Example 2 – Explosive AR(1) Model

STATA code

set obs 500 /*set number of observations*/

gen time=_n /*create time trend*/

gen y=0 if time==1 /* first observation set y=0*/

gen e=rnormal(0, 1) /*create a random number*/

replace y=(1.16*y[_n-1])+e if time~=1 /*AR(1) model =1.16*/ twoway (line y time) /*line plot*/

Example 3 – Non-stationary AR(1) Model

STATA code

set obs 500 /*set number of observations*/

gen time=_n /*create time trend*/

gen y=0 if time==1 /* first observation set y=0*/ gen e=rnormal(0, 1) /*create a random number*/ replace y=y[_n-1]+e if time~=1 /*AR(1) model =1*/ twoway (line x time) /*line plot*/ Notice

t

y is not mean reverting. Random walk =1

In the model:

1t t t y y

if 1 then t y is said to contain a UNIT ROOT i.e. is non-stationary

So 1t t t y y subtract 1t y from the LHS and RHS:

111t t t t t y y y y

t t y and because t is white noise t y is a stationary series.

Example 3 (continued) – Non-stationary AR(1) Model and First Difference

- A series t y is integrated of order one, i.e. t y I (1), and contains a unit root if t y is non-stationary but t y is a stationary series.

- Possible that the series t y needs to be differenced more than once to achieve a stationary process.

- A series t y is integrated of order d , i.e. t y I (d) if t y is non-stationary but d t y is a stationary series: Note: 21

1

t t t t t t t y y y y y y y

2.S PURIOUS REGRESSION

Why worry whether t y is stationary?

Most macroeconomic time series are trended and in most cases non-stationary processes.

Using OLS to model non-stationary data can lead to problems and incorrect conclusions.

a.high R squared often >0.95

b.high t values

c.theoretically variables in the analysis have no interrelationship Why does non-stationarity arise in macro data?