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I cannot directly answer your question, but I think I can shed some light. What I seem to show is that under some restrictions, the unconditional variance is finite. However, I am not sure how to relate my answer to the notion of volatility clustering. Note that if we assume stationarity, then $\sigma^2=E(\epsilon_t^2)$ for all $t$ and it follows by the law ...


3

Sorry for any confusion in my previous answer but there are 2 steps to this process 1) tracing out the impact on the $\Delta$y's and the lags of the shocks and then 2) accumulating the shocks to get the cumulative response (Note that my original answer tried to do this simultaneously). So in the first step (ignoring the intercept, the dummies and the ...


2

Plot things. Do the raw data look like they have cycles? (I assume so, given that you are talking about economic data and want to use an autoregressive model.) Next, try to get the order of the autoregressive model by taking differences until the cycles disappear. This can also be tested formally using (a version of) the Durbin-Watson test. There will likely ...


2

Ok. I made some small computation errors and got confused here. The notes make sense with the following notational assumptions. If I write out the AR(1) process as follows (ignoring drift) $$ r_{t+1} = r_t + \epsilon_{t+1}, $$ then we have $\text{Cov}(r_t, r_{t+j}) = \frac{\rho^j}{1 - \rho^2} \sigma_\epsilon^2$, where $\sigma^2_\epsilon := Var(\epsilon)$. ...


2

For an AR(1) process (I omit any drift), the coefficient on the lag is the 1st-order correlation coefficient, $$r_{t+1} = \rho r_t + u_{t+1}$$ So $$r_{t,t+2} = \sum_{i=1}^2 r_{t+i} = r_{t+1} + r_{t+2} = \rho r_t + u_{t+1} + \rho r_{t+1} + u_{t+2} $$ $$=\rho r_t + u_{t+1} + \rho \big(\rho r_t + u_{t+1}\big) + u_{t+2} = \rho(1+\rho)r_t+(1+\rho)u_{t+1} + ...


2

I don't know R code but are you estimating an intercept in the ARIMA(0,1,0) model? Because if not, this could be why there is a difference, since you are estimating an intercept in the ARIMA(0,0,0) model.


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The weight matrix that you have illustrated needs to be row-normalized, i.e. the row-sum must be one in order for the rho parameter to have the proper parameter bounds, i.e. between (-1/lambda,1), where lambda is the smallest real eigenvalue of the W matrix. When the weight matrix is row normalized and there is a change in the dependent variable, the shock ...


1

(Before applying serial correlation tests to the residuals, you want to visually inspect the residuals for whiteness---look at the sample ACF and PACF. Serial correlation test statistics are often some kind of transformations of the sample ACF.) ...in AR(p) model..The error term must be i.i.d... That statement is not correct. The i.i.d. condition is not ...


1

Yes you can use Breusch–Godfrey (BG) test for autocorrelation also in AR(p) models and dynamic models in general (see Verbeek Guide to Modern Econometrics where BG is applied to dynamic models in some examples - one of such examples is on page 142 in the 4th ed). As a matter of fact BG test is, generally speaking, the preferred test for autocorrelation in ...


1

I think that the wikipedia steps are not completely correct. When you perform Johansen cointegration test you first have to pretest the data to find if they have the same order of integration. That is all variables have to be either $I(1)$ or $I(2)$ etc. (see Verbeek, 2008). There are some cointegration tests and models that relax this assumption but ...


1

Intuitively, you test for cointegration because if two variables are cointegrated, they represent only a "one dimensional" family of data points - even if you have a million data points from that sample, they will all fall close to the same subspace, and in general that will mean that you will have many values of parameters for your regression problem which ...


1

Stock and Watson use log approximation and quarterly data. Premultiplying the log approximation by 400 could be due to data being quarterly. Your second method is fine too.


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I believe that @Andrew_M is right, this is caused by differences in default options in the implementations of ARIMA across statistical applications. library(forecast) births <- scan("http://robjhyndman.com/tsdldata/data/nybirths.dat") birthstimeseries <- ts(births, frequency=12, start=c(1946,1)) level010 <- arima(births, order = c(0,1,0), include....


1

I think you can use VAR models in these kind of applied policy issues. I think you are interested by some questions like "how about the effect of exports on GDP and GDP on export ?" So, if it is the case, I think the most appropriate way is to use VAR models in which you can analyze the causality (like Granger Causality) between your key varibles. But you ...


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