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Consider two credits D1 and Dz Assume that the survival of the credit is dependent on a latent variable A;~N(O, 1). Specifically, a default occurs before time Tif t...

Question

Consider two credits D1 and Dz Assume that the survival of the credit is dependent on a latent variable A;~N(O, 1). Specifically, a default occurs before time Tif the value of A; is less than a time dependent threshold C; (T) ie-,P(t <T) = P(A; < C(T)) =1-Q (t)Here, Q;(T) is the time T survival probability of credit implied by the CDS market: Consider the following _ one-factor model:Ai = B+z + I1 - Bzewhere Z~N(O,1) is the market factor and &~N(O,1) are the idiosyncratic factors spec

Consider two credits D1 and Dz Assume that the survival of the credit is dependent on a latent variable A;~N(O, 1). Specifically, a default occurs before time Tif the value of A; is less than a time dependent threshold C; (T) ie-, P(t <T) = P(A; < C(T)) =1-Q (t) Here, Q;(T) is the time T survival probability of credit implied by the CDS market: Consider the following _ one-factor model: Ai = B+z + I1 - Bze where Z~N(O,1) is the market factor and &~N(O,1) are the idiosyncratic factors specific to each credit: Within this framework; there are no explicit = spread dynamics. The only events that we can observe are the defaults of credits_ A credit can have spread dynamics when we condition on the default and survival behavior of the other credits to which this credit is correlated, .If Di defaults at future time T1 and it has non-zero asset correlation with Dz, then we would expect this default to have an impact on the conditional default probability of Dz. We denote the default time for credit B as Tz and the correlation between credits as p Calculate Q2(t,T) the forward survival curve for a credit Dz conditional on default of credit Di at time and compare Qz(t,T) with Qz(t,T) the unconditional forward survival curve_ Please comment on the relationship. Assume that the initial CDS spread for credit D is flat at 60 bps, the CDS spread for credit Dz is flat at 120 bps and the recovery rate is R = 40% for both credits. Calculate the forward spread curve of credit Dz conditional on default of credit D at time t e(0,5). Plot the calculated forward spreads for correlation between assets P € {-0.5,-0.25,0,0.25,0.5}. Discuss the relationship between the conditional spread curve of credit Dz and the asset correlation p_



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This question asks you to study the so-called Beveridge Curve from the perspective of cointegration analysis. The U.S. monthly data from December 2000 through February 2012 are in BEVERIDGE.RAW.
(i) Test for a unit root in urate using the usual Dickey-Fuller test (with a constant) and the augmented DF with two lags of curate. What do you conclude? Are the lags of curate in the augmented DF test statistically significant? Does it matter to the outcome of the unit root test?
(ii) Repeat part (i) but with the vacancy rate, vrate.
(iii) Assuming that urate and vrate are both I(1), the Beveridge curve,
$$u r a t e_{t}=\alpha+\beta vrate +u_{t}$$
only makes sense if urate and vrate are cointegrated (with cointegrating parameter $\beta<0 )$ . Test for cointegration using the Engle-Granger test with no lags. Are urate and vrate cointegrated at
the 10$\%$ significance level? What about at the 5$\%$ level?
(iv) Obtain the leads and lags estimator with $cvrate_{t}$, $cvrate_{t-1}$ and $cvrate_{t+1}$ as the I(O) explanatory variables added to the equation in part (ii). Obtain the Newey- West standard error for $\hat{\beta}$ using four lags $(\mathrm{so} g=4$ in the notation of Section 12.5$) .$ What is the resulting 95$\%$ confidence interval for $\beta$ How does it compare with the confidence interval that is not robust to serial correlation (or heteroskedasticity)?
(v) Redo the Engle-Granger test but with two lags in the augmented DF regression. What happens? What do you conclude about the robustness of the claim that urate and vrate are cointegrated?

First one. The test for a unit root in series you rate unemployment rate using the usual dickey fuller test with a constant yeah. And the augmented dickey fuller with two legs of change of unemployment rate. I find that seven both times we are unable to reject the now hypothesis that unemployment rate series is a unit fruit. The legs are not significant. However, the significance of the legs matters. So the outcome of the unit root test, we will repeat what we have done in part one two series vacancy rate and report the result in part two. I guess similar result. So the rate is a unit root. Well part one and two. I use package the R. Package A. T. S. A. And the function is a D. F. Dot test. R. Three. We assuming that unemployment rate and vacation re rate are both integrated of level one. We test for co integration using the angle grandeur test with no legs. So the step the steps are as follow. We first regress, you read on the rate then we yet the residual and we run the key fuller has on the residual to see whether the residuals our unit root. I find that you're right and we rate Arco integrated at the 5% level. Yeah Heart Forest. I get the leads and lacks estimator of the change in vacancy rate and I did note that uh CB rates up minus one. This is for the lack and plus one is for the lead. This is a regression result. So the usual centered errors are in green and in round brackets, the robots that Iran's are in blue and in square brackets you can see that the main estimate on vacancy rate is highly significant. This one is not correct. So the centered errol the usual one for the estimate of the first lack of change in vacancy rate is 164 In all cases except for the estimate of the lead of C. V. Right. The robust standard Iran's are larger than the usual standard errors. This is usually the case it happens but rare that the robot standard errors are smaller than the usual standard errors. The r square of this regression is 0.77 So for the rate, because the robot standard error is larger than the usual standard error. So we will get a wider confidence interval if we use a robot standard error and for confidence interval you will run this function in our count in and you impose the name of the regression. It was spits all the 95% confidence intervals for all explanatory variables. The default version is the 95% interval. But because the standard barrel of this estimate is are very close, two versions are very close to each other so the confidence intervals should be roughly equal. Yeah. Last part. What you could say about real business of the claim that you rate and the rate are co integrated. Yeah. When I run the test and good grandeur, the results are not consistent across alternative types of process. In one case I can reject the notion that the residuals are united and for all the cases I cannot reject. So I conclude that the claim that you rate and be rate our co integrated is not robust.


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