clear
set mem 15m

use http://www.pabsta.qc.ca/sites/default/files/tsx.dta

cd "/Users/pabsta/documents/2-enseignement/ECON452/tutorial5/"

log using "OUTPUT/tutorial5-2.txt", t replace

tsset date

tsline tsx

/* Exponential increase of the process: 
- This is not stationarty! 
- This also suggest that logarithms can linearize the sytem
*/

gen ltsx = ln(tsx)

tsline ltsx
// Trend or drift? 
dfuller ltsx 
dfuller ltsx, trend
dfuller ltsx, drift

/*
There seems to have a unit-root. And this unit-root is gone after differenciating. 
Note that adding a trend does not remove the unit-root. 

We first need to get rid of the time trend. 
*/

gen DlogGDP = D.logGDP

dfuller DlogGDP
dfuller DlogGDP, trend
dfuller DlogGDP, drift
//This seems to do the trick: no unit-root anymore. 

//Next move: estimate the AR? and MA(q) components
pac DlogGDP
//First and fifth lags are important. Looks like an AR(1). 

ac DlogGDP
//High persistence of the series, "almost" no exponential decline. 
//Some oscillating patterns, which indicate that the AR component dominates the picture. 
//The first 10 lags are significantly different from zero.
//We proceed to a first broad estimation

//Programming tip: the command below lets the estimation run by itself
//without having to type everytime the screen displays "more".
set more off


arima DlogGDP, ar(1,2,3,4,5) ma(1,2,3,4,5,6,7,8,9,10)
//Note that it takes a while to estimate!
//Some unestimated coefficients: common factors. 
//Lets try with smaller values for the MA(q)
arima DlogGDP, ar(1,2,3,4,5) ma(1,2,3,4,5,6,7,8)
//Still problems of unestimated coefficients. 
//Reduce again.
//...
arima DlogGDP, ar(1,2,3,4,5) ma(1,2,3,4)
//Test if the last lags of the MA are insignificant
test ([ARMA]L4.ma = 0) ([ARMA]L3.ma = 0) ([ARMA]L2.ma = 0) ([ARMA]L1.ma = 0)
//Too much information (reject H0)

//The last two.
test ([ARMA]L4.ma = 0)  ([ARMA]L3.ma = 0)
//New estimation

arima DlogGDP, ar(1,2,3,4,5) ma(1,2,3)
//Negative variance?!? (look at the confidence region! This is clearly wrong)
arima DlogGDP, ar(1,2,3,4,5) ma(1, 2)

test ([ARMA]L1.ma = 0) ([ARMA]L2.ma = 0) 
//Reject H0

arima DlogGDP, ar(1,2,3,4,5) ma(1)
test ([ARMA]L4.ar = 0) ([ARMA]L5.ar = 0)
test ([ARMA]L4.ar = 0) ([ARMA]L5.ar = 0) ([ARMA]L3.ar=0)

arima DlogGDP, ar(1,2,3) ma(1)
//Further testsÉ
test ([ARMA]L3.ar = 0) ([ARMA]L2.ar = 0) ([ARMA]L1.ma=0)
test ([ARMA]L3.ar = 0) ([ARMA]L1.ma=0)
arima DlogGDP, ar(1,2) ma(1)
arima DlogGDP, ar(1,2)
arima DlogGDP, ar(1)
//One spec to keep in mind. 
//Now what if we reduce AR firsts? 
arima DlogGDP, ar(1,2,3,4,5) ma(1,2,3,4)
test ([ARMA]L5.ar = 0) ([ARMA]L4.ar = 0)

//Not significant at the 95% level but significant at 10%É
arima DlogGDP, ar(1,2,3, 4) ma(1,2,3,4)
//Common factor
arima DlogGDP, ar(1,2,3) ma(1,2,3)
test ([ARMA]L3.ar = 0) ([ARMA]L2.ar = 0)
//reject hO
arima DlogGDP, ar(1,2) ma(1,2,3)
//First lag significant at 10%, but nothing elseÉ 
arima DlogGDP, ar(1) ma(1,2,3)
arima DlogGDP, ar(1) ma(1,2)
//This is a candidate model. 

//Which one should we pick? A criterion is the one with the smallest variance, namely the last one. 
varsoc DlogGDP
//This suggest as well that we should have an AR(1)
predict DlogGDP_pred, xb

//Interpretation for this: last year's growth is a good predictor of this year's growth. 
//However, the past innovations in growth have persistence (2 years).


log close