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Bivariate (multivariate) analysis. Aim: to estimate the model: Y t =b*X t + N t ; b= the regression coefficient expressing the effect of X on Y N t =noise term (error term), including other causes of Y besides X.

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bivariate multivariate analysis

Bivariate (multivariate) analysis

Aim: to estimate the model:

Yt=b*Xt + Nt;

b= the regression coefficient expressing the effect of X on Y

Nt=noise term (error term), including other causes of Y besides X

slide2
Is the series stationary? If not: difference the series: Xt = Xt-Xt-1

The differencing reduces the risk of spurious correlations, since an omitted variable is more likely to be correlated with the explanatory variable due to common trends than as a result of synchronization in the yearly changes.

Plot (scattergram) Yt against Xt to detect outliers

slide3

Are the series stationary? If not: difference both X and Y; Xt = Xt-Xt-1Plot (scattergram) Yt against Xt to detect outliers. Remedies: shorten series; dummy variable

slide4
Estimate the cross-correlations (CCF) between the pre-whitened Xt and the pre-whitened Yt in order to detect lag-structure (separate lecture on CCF, now: assume no lag-structure)
  • Specify functional form (more details later):
    • Linear
    • Semi-logarithmic
    • Log-log
slide5
Yt=b*Xt + Nt; The noise (Nt) is allowed to

have a temporal structure that is modelled

through AR and/or MA-parameters:

Nt= Nt-1+ et. The residuals et= white noise.

If the residuals ≠ white noise (but

autocorrelated), the estimate of b will be

unbiased, but the SE of b will be

underestimatedfalse significance.

Thus, we must identify and estimate a model

for Nt

slide6
1. Estimate the model:

Yt=b*Xt + Nt ;don’t include any noise parameters.

2. Identify the structure of the noise term on the basis of the ACF and PACF of the residuals.

3. Re-estimate the model including noise parameters.

4. Diagnostic test of the model: are the residuals white noise. If not: modify the noise parameters on the basis of the ACF and PACF of the residuals, and re-estimate the model.

slide7
Example: Assaultratet=b* Alkuttott + Nt

Assaultrate=assaults/100,000

Alkuttot=public (bars etc) alcohol consumption

1. twoway (line alkuttot year) if year>1955

2. twoway (line assaultrate year) if year>1955

/Trending: should difference, but analyse raw data as an exercise /

3. generate alkutdif=d.alkuttot

4. generate assaultdif=d.assaultrate

5. twoway (scatter assaultdif alkutdif)

6. corrgram assaultdif, lags(20)

7. corrgram alkutdif, lags(20)

8.arima alkutdif, arima(1,0,0)

9.predict resass, r

10. corrgram resass, lags(20)

slide8
11. xcorr resass assaultdif, table lags(10)

12. arima assaultrate alkuttot, arima(0,0,0)

13. predict resmod1, r

14. arima assaultrate alkuttot, arima(1,0,0)

15. predict resmod2, r

16. corrgram resmod2, lags(20)

17. arima assaultrate alkuttot, arima(0,1,0)

18. predict resmod3, r

19. corrgram resmod3, lags(20)