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ECON5022_MAIN_ 2020

S ec t i on A

1. Consider the following ARIMA(0,1,1) process

∆ xt = et + 9eto1 , where 0 < I9I < 1

and {et} is a white noise with mean zero and variance 72 .

1.1. Derive the Beveridge-Nelson decomposition. You should ﬁnd that

t

xt = (1 + 9) es - 9et + (9e0 + x0 ) .

s=1

Show your calculations.

[20%] 1.2. Compute the h-steps-ahead forecast for xt .

[10%] 1.3. Show that the forecast mean squared error can be written as ┌ 1 + (h - 1)(1 + 9)2 ┐ 72

[20%]

2. Let xt = 7tZt be a GARCH(1,1), where {Zt} is a sequence of independent standard normal random variables, and

7t(2) = u + axto(2)1 + 87t(2)o1 .

2.1. Show that the h-step-ahead forecast 7t(2)+hlt of 7t(2) can be written as

ho2

(a + 8)j + (a + 8)ho1 7t(2)+1

j=0

(1)

[15%]

2.2. Assume (a + 8) < 1. Use the ARMA representation for xt(2) to compute 72 , the unconditional variance of xt .

Show that equation (1) can be rewritten as

7t(2)+hlt = 72 + (a + 8)ho1 (7t(2)+1 - 72 )

Argue that 7t(2)+hlt converges to 72 as h → o.

[20%]

2.3. Modify appropriately the above GARCH(1,1) to obtain a TGARCH (Threshold

GARCH) . Compute the News Impact Curve and discuss brieﬂy how the TGARCH you derived accounts for the leverage eﬀect.

[15%]

S ec t i on B

3. If the ﬁnancial market is eﬃcient, the spot and future prices should never drift too far apart, suggesting that a cointegrating relationship might be appropriate. Brooks, Rew and Ritson (2001) examines the lead-lag relationship between the FTSE 100 future contract (Ft) and its underlying asset, the FTSE 100 stock index (st) using 10-minutes observations for the period June 1996-1997.

Table 1 in Fig 1 reports the descriptive statistic and the Dickey-Fuller (DF) test for the two series. ln(.) denotes the natural logarithm. The returns tt and ft in Panel C are deﬁned as tt = ∆ ln st and ft = ∆ ln Ft .

Table 2 in Fig 1 display the results for the cointegration analysis. Panel A reports the results for the regression

ln st = y0 + y1 ln Ft + Zt ,

whereas the residual-based test for cointegration is shown in Panel B. Panel C presents the estimates for the Error Correction Model (ECM)

∆ ln st = 80 + 6 to1 + 81 ∆ ln sto1 + a1 ∆ ln Fto1 + 3t ,

(2)

where 3t is an error term, and t = ln st - 0 - 1 ln Ft. Table 3 report the results for the ECM including the cash and carry cost (ECM-COC), obtained replacing t on (2)

Figure 1: Stationarity and Cointegraton Analysis

with

t(COC) = ln st - 0 - 1 ln Ft - 2 (s - d)(T - u).

where T is the maturity date of the future contract, s and d are the continuously com- pounded risk free rate and dividend yield, respectively.

Finally, the out of sample one-step-ahead forecasts for the ECMs have been computed (1040 10-minutes observations). The forecasts are then compared to the actual returns, with the forecast accuracies evaluated on the standard statistical criteria of root mean squared error (RMSE), mean absolute error (MAE), and the percentage of cases where the forecasts predict correctly the direction of movements of the spot index. The com- parison results are reported in Table 4 (see Fig 2). The forecast from an ARMA model for tt have been included as a benchmark (ARIMA)

Figure 2: Comparison of out of sample forecasting accuracy

3.1. Are the series ln st and ln Ft stationary? Are the returns stationary? Justify your

answers.

[12.5%]

3.2. Explain brieﬂy the residual-based approach to test for cointegraton. Are ln st and ln Ft cointegrated? Justify your answer.

[12.5%]

3.3. Is there an empirical evidence that the future market does lead the spot market?

Justify your answer using the estimation output in Fig 1, Table 2, Panel C. Comment on all the coeﬃcients.

[12.5%]

3.4. Which model is the best performer in the out-of sample forecast? Justify your

answer. Why are the direction forecasts of particularly interest in this application? [12.5%]

The relevant critical values can be found in Appendix A.

4. In an application to daily exchange returns, Christoﬀersen (1998) evaluated the interval forecast from J.P. Morgan (1995), along with two competing forecasts. Let et the one step-ahead forecast error. The interval forecast for et suggested by J.P Morgan is,

[Ltlto1 (p), Utlto1 (p)] = ┌ -(1 − ← )/2 7t , -(1+ ← )/2 7t ┐

(3)

o

7t(2) = (1 - A) Ajet(2)oj = A7t(2)o1 + (1 - A)et(2)o1 , A = 0.94. j=0

and - α satisﬁes Pr(Z < -α) = a. Z is assumed to have standard normal distribution.

Consistent with most of the literature on exchange rate prediction, J.P. Morgan does not model any conditional mean dynamics in its forecast. The time-varying interval in (3) is simply placed around a constant mean.

The ﬁrst competitor is a GARCH(1,1) with Student’s t innovation (GARCH(1,1)-t). The second is a conﬁdence interval based on the in-sample estimated unconditional distribu- tion (static forecast).

In this exercise we focus on the daily log-diﬀerences in the British Pound vis-a-vis the U.S. Dollar. A total of 4,000 observation (January 1980- May 1995) is used; the ﬁrst half of the sample for parameter estimation, the second half for out-of-sample interval forecasting and forecast evaluation. The results are presented in Fig 3.

The top panel shows the LR statistics of conditional coverage for three interval forecast. The long dash is J.P. Morgan’s Risk-Metrics forecast, the solid line the GARCH(1,1) -t forecast, and the short dash is the static forecast. The solid horizontal line represents the 5 percent signiﬁcance level of the appropriate x2 (k) distribution (3.84 and 5.99 for k = 1 and k = 2 degrees of freedom, respectively).

The test values are plotted for coverages p ranging between 50 and 95 per cent. The middle and bottom panels show the corresponding values of the LR tests of unconditional coverage and independence, respectively.

4.1. State the null and the alternative hypothesis for the LR test of independence and

the LR test for unconditional coverage.

[20%]

4.2. Explain why it is important to test for conditional coverage.

[15%]

4.3. Comment on the results displayed in Fig. 3.

[15%]

Appendix B reports some formulas about the LR tests.

Figure 3: Likelihood ratio statistics of conditional coverage (Lcc), unconditional coverage (Lud), and independence (Lind)

A pp e nd i x A

Figure 4: Critical values DF test.

Figure 5: Critical Values Residual-based Unit Root Test.

A pp e nd i x B

Consider the sequence of intervals { ┌ Ltlto1 (p), Utlto1 (p) ┐ } with coverage probability p, for the time series {xt}, u = 1, . . . n. Deﬁne the indicator variable

1, if xt e [Ltlto 1 (p), Utlto 1 (p)]

It = ．

, 0, if xt [Ltlto1 (p), Utlto1 (p)]

(a) The LR test of Unconditional Coverage (uc) is formulated as

LRuc = -2 log[L(p; I1, . . . , In)/L( ; I1, . . . , In) ~ x2 (1)

where

L(p; I1, . . . , In) = (1 - p)n0 pn1 , L(m; I1, . . . , In) = (1 - m)n0 mn1

are the likelihoods under the null and under the alternative, respectively. ni denotes the number of observations with value It = i, i = 0, 1, and is the maximum likelihood estimator of m .

(b) The LR test of independence (ind) is formulated as

LRind = -2 log[L( 2 ; I1, . . . , In)/L( 1 ; I1, . . . , In) ~ x2 (1)

where 2 and 1 are estimators of the transition probability matrix under the null (Π2 ) and the alternative (Π 1 ), respectively, with

Π 1 = ┐ , Π 2 = ┐ ,

mij = Pr(It = jIIto1 = i) and

L(Π1 ; I1, . . . , In) = (1 - m01 )n00 m0(n)1(01) (1 - m11 )n10 m L( Π 2 ; I1, . . . , In) = (1 - m2 )(n00 +n10 )m

(c) The LR test of Conditional Coverage (cc) is formulated as

LRcc = 2 ln[L(p; I1, . . . , In)/L( 1 ; I1, . . . , In)] = Luc + Lind ~ x2 (2).