Group Assignment for Quantitative Analysis

2018S1 Quantitative Analysis Group Assignment

Question 2

Part (a)

Unit Root Test

Time series rvt is covariance stationary if

1. E(rvt) does not depend on t
2. Var(rvt) exists and does not depend on t
3. COV(rvt, rvt-j) does not depend on t for every j

In order to test the stationarity of rvt, we need to conduct a unit root test.

Step 1:

![pic 1]

∆𝑦!  = 𝛼!𝑦!!! + 𝛿! +        𝛾!∆𝑦!!! + 𝜖!

!!!

𝐻!: 𝛼!  = 0        𝑈𝑛𝑖𝑡 𝑅𝑜𝑜𝑡[pic 2][pic 3]

𝐻!: 𝛼!  < 0   𝑁𝑜 𝑢𝑛𝑖𝑡 𝑟𝑜𝑜𝑡[pic 4][pic 5]

Step 2:

Conduct a unit root test for RV, with test for unit root in level, including an intercept in test equation and using modified Akaike automatic selection with default maximum lags of 34.

[pic 6]

𝑡 − 𝑠𝑡𝑎𝑡  𝜏   =        𝛼[pic 7][pic 8][pic 9][pic 10][pic 11]

= −8.6264

Step 3:

The value of test statistic is -8.6264. The critical value at 5% significance level is -2.8617.

Step 4:

Since -8.6264<-2.8617, we should reject the null hypothesis at 5% significance level. The p-value gave us the same result. P-value=0 which is less than 5%, we reject null hypothesis of unit root at 5% significance level, favouring that RV has no unit root. That is, RV, is I (0), not I (1).

Estimation of unrestricted AR (22)[pic 12]

The estimated model is:

𝑟𝑣! = 4.6351 + 𝑢![pic 13]

𝑢!  = 0.1082𝑢!!! + 0.0653𝑢!!! + 0.1988𝑢!!! − 0.0147𝑢!!! − 0.0175𝑢!!! + 0.3685𝑢!!![pic 14][pic 15][pic 16][pic 17][pic 18][pic 19][pic 20]

− 0.0347𝑢!!! − 0.0085𝑢!!! − 0.0965𝑢!!! + 0.0352𝑢!!!" + 0.0213𝑢!!!![pic 21][pic 22][pic 23][pic 24][pic 25]

− 0.1271𝑢!!!" + 0.0203𝑢!!!" + 0.0178𝑢!!!" + 0.0609𝑢!!!" − 0.0130𝑢!!!"[pic 26][pic 27][pic 28][pic 29][pic 30]

− 0.0037𝑢!!!" + 0.0487𝑢!!!" + 0.0149𝑢!!!" + 0.0275𝑢!!!" + 0.0038𝑢!!!"[pic 31][pic 32][pic 33][pic 34][pic 35]

+ 0.0204𝑢!!!![pic 36]

It is noteworthy that lags 4, 5, 8, 11, 13, 14, 16, 17, 19, 21, 22 are insignificant (p-value < 0.05). However, we would still stick to AR (22) model, as specified in the question.

According to adjusted R-squared, AR (22) model is able to explain 21.6636% of the variations in realised variance during period t.

[pic 37]

There is no significant autocorrelation or partial autocorrelation, this matches with the characteristic of a white noise process. Given the sample size of 6473 observations, the significance of the autocorrelations  and  partial  autocorrelations  can  be  gauged  by  using  the  5%  significance level

critical values of ±   !.!"[pic 38]

!"#$

= ±0.0244.

Estimation of HAR model

[pic 39]

The estimated model is:

𝑟𝑣!  = 1.5752 + 0.0117𝑟𝑣!!! + 0.2895𝑟𝑣!!!,!!! + 0.3589𝑟𝑣!!!!,!!! + 𝜖!

The p-value for RV (-1) is 0.4219, which is greater than 5% significance level (0.05), it indicates this variable is insignificant. As for other parameters, their p-values are all less than 0.05, therefore, these parameters are significant.

According to adjusted R-squared, HAR model is able to explain 8.8062% of the variations in realised variance during period t.

[pic 40]

The autocorrelations and partial correlations show significant spikes at lags 2, 3, 4, 5, 6, 7 and some other lags afterwards. The first lag is insignificant as its P-value = 0.17 which is larger than 0.05, Given the sample size of 6473 observations, the significance of the autocorrelations and partial

autocorrelations  can  be  gauged  by  using  the  5%  significance  level  critical  values  of  ±   !.!"   =[pic 41]

!"#$

±0.0244.

We compare the two models in terms of the following aspects:

• Adjusted R-squared: AR (22) model is able to explain 21.6636% of the variations in RV while HAR model can only explain 8.8062% of the variations.
• Information criteria: AR (22) model has both lower Akaike info criterion and Schewarz criterion (8.0279 and 8.0520), comparing with HAR model’s figures (8.1770 and 8.1811).

Therefore, AR (22) model performed better when estimating the results and should be adopted.

Part (b)

1. AR (22) model one-day-ahead variance forecasts

50[pic 42][pic 43][pic 44][pic 45][pic 46][pic 47]

40

30

20

10

0

-10

-20

-30

I        II        III        IV        I        II        III        IV        I        II        III        IV

2013        2014        2015

[pic 48]

1. AR (22) model graph of actual values against forecasts

50[pic 49][pic 50][pic 51]

...