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6

I don't have advice specific to error correcting model (ECM) setting, but in undergraduate applied econometric class they gave us the generic advice to continue to extend lags in the model until the residuals of the fitted model were serially uncorrelated. For example, in the US life expectancy data, residuals of male life expectancy is serially uncorrelated ...


6

"Who moves first" can be conveniently detached from any causal inference, since there may be some third variable influencing both. Certainly one could build a logical and reasonable argument that links increases in investment to increases in GDP (or vice versa), but it is not necessary. Then, examining cross-correlation coefficients on the levels or on an ...


6

First, it is important to note that there is a substantial literature both developing and using nonlinear VAR estimation (for example, see papers here, here, here, and here). The reasons linear VARs are seen a lot, though, are similar to the reasons that least squares (in its varying forms) is seen a lot. The model is very transparent and analytically ...


5

Morten O. Ravn and Harald Uhlig (2002) This paper complements these insights using two different analytical approaches. The first approach uses the time domain and focuses on the ratio of the variance of the cyclical component to the variance of the second difference of the trend component: this ratio is often used for calculating the ...


5

To get Quah and Vahey's method, you should probably get a good handle on the Wold Representation Theorem first. Christiano has good notes on it here. Then, you can apply it to VAR. There's a pretty good basic explanation in Greene's Econometric Analysis. In the Sixth Edition, it's in Chapter 20 (Models with Lagged Variables) starting with Section 20.6, "...


5

To show that transformation is measure preserving you need to show that full preimage $S^{-1}(A)$ of any set A in the Borel $\sigma$-field on [0,1) is again in the same $\sigma$-field, i.e. it is measurable, and that $Pr\{S^{-1}(A)\} = Pr\{A\}.$ As you have correctly pointed out, it will suffice to show it for any base of usual topology, namely, all open ...


5

An I(1) series is also known as a series with a unit root. Therefore, the econometric tests to inquire on the order of integration of a time-series are referred to as "unit-root tests". There are three widely used unit-root tests: Augmented Dickey-Fuller (ADF), Phillips-Perron (PP) and Kwiatkowski–Phillips–Schmidt–Shin (KPSS). The null hypothesis (H0) in ...


4

This construction you describe is not fully general. In fact it characterizes strictly stationary time series. You see that it's shift-invariant. This operator $S$ is essentially a shift operator. For comparison, here's the usual definition of, let's say discrete-time, processes: Definition A stochastic process is a sequence $\{ X_t \}$ of Borel measurable ...


4

The terms "one-sided" and "two-sided" lag-polynomials, are used when the text considers the option to specify an equation with both "lags and leads", i.e a relation where both past but also future values co-vary with current value. When the lag-operator is "one-sided" it contains only lags in the one direction. When it is two-sided, it contains "lags" in ...


4

Granger Causality is exactly what you are looking for. Don't be fooled: Granger Causality does not imply causality. To say that $X$ Granger-causes $Y$ merely means that lagged values of $X$ add some predictive power when predicting $Y$ as compared to a univariate autoregression of $Y$.


4

I think it is a very classical economic teaching problem - showing how something is relevant in the real world. First, it solved a problem where linear regressions could lead to spurious results: https://en.wikipedia.org/wiki/Cointegration ...Before the 1980s many economists used linear regressions on (de-trended[citation needed]) non-stationary time ...


4

Economists (most of them) build their models assuming most of the time stochastic dynamic equilibrium. So Economics does not contrast "dynamic" with "equilibrium" - it synthesizes them. It is stochastic in the sense that random shocks are acknowledged. It is dynamic in the sense that it may revolve around a deterministic or stochastic trend. And it is an ...


3

We have the recurrence relation $$x_{k+1} = \frac{x_{k+8}}{x_{k+1}}$$ If the denominator is nonzero, this recurrence relation can be rewritten as follows $$x_{k+7} = x_k^2$$ Assuming positivity and taking the logarithm of both sides, we obtain a linear recurrence relation $$\ln (x_{k+7}) = 2 \ln (x_k)$$ Shifting, $$\ln (x_{k+1}) = 2 \ln (x_{k-6})$$ ...


3

Does it change any interpretation of the elasticity ($β_1$)? You walk into the firm where you work as an analyst, and the Sales Director calls and asks "I want to raise the price $10\%$ today. How will demand be affected in percentage terms? Well, you don't expect income to have changed from one day to the next, and what was demanded yesterday -it was ...


3

In your situation (after reading the comments) I would use the 10 year data over the 2 year company specific data, unless you can identify some reason why your specific company is especially atypical for the industry. Considering the length of business cycles we typically see in the developed world, about 5 years, any 2 year period may not be reflective of ...


3

The derivation of the score process is correct. To verify that the process is a Martingale, recall the definition. It becomes clear that if we substitute $W_{t+1}$ back into the equation $$ s_t(\theta \mid \textbf X) = \begin{bmatrix} (1 - \beta_0) \sum_{j=1}^t W_j \\ \sum_{j=1}^t W_j (X_{j-1} - \alpha_0) \end{bmatrix}. $$ Because $W_{t+1}$ are Normal with ...


3

The paper Cointegrating Regressions with Messy Regressors: Missingness, Mixed Frequency, and Measurement Error (J. Isaac Miller (2009)) seems to have what you are looking for. We consider a cointegrating regression in which the integrated regressors are messy in the sense that they contain data that may be mismeasured, missing, observed at mixed ...


3

The critical points here are the phrases "it is recommended that...", and "using four lags is a norm". "Recommended" and "norm" based on what? On a specific prevailing theoretical model, or on past experience with the specific kind of data? If it is the latter, you do not conflict with any economic theory, if you try something different. This is also a good ...


3

If anyone per chance runs into the same issue, The reason the MLE is exploding is that a system with explosive roots (eigenvalues outside the unit circle), might make the Kalman-filter predictions explode at a point, and thus creating a few very large MLE values. The solution is to rig the MLE function s.t. if any of the eigenvalues lie outside the unit ...


3

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

For a second-order stationary series it is the correlation coefficient between the dependent value and its lag. Specify $$y_{t+1} = a+ \beta y_t + u_{t+1}\qquad u_{t+1}= \text{white noise}$$ The correlation coefficient between $y_{t+1}$ and $y_{t}$ is defined as usual $$\rho_{(1)} = \frac{\text{Cov}(y_{t+1},y_{t})}{\sigma(y_{t+1})\sigma(y_t)}$$ $$\text{...


2

You could do a binomial-tree approximation to the process for $z_t$ and then have a different process control the number of steps you take on the tree. This preserves recombining property and it is essentially the method explored in On the Computation of Continuous Time Option Prices Using Discrete Approximations (Amin (1991)) We develop a class of ...


2

Let the $n$ states of the finite-state markov chain be denoted by $\{x_1,...,x_n\}$ and let $\vec e = [e(x_1), ..., e(x_n)]'$. Now, first note that because $X_{t+1} \mid X_t$ is independent of $W_{t+1}$, we can write \begin{align*} \exp(\eta) e(x) &= E[\exp(D'x + x' F W_{t+1})] E[ e(X_{t+1}) \mid X_t = x] \\ &= \exp(D'x + x' F F' x)\, E[ e(X_{t+1}) ...


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

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

The OP is examining a deterministic trend, and whether it stops being present after a point in time. So the model could be something like $$y_t = \alpha t + \beta y_{t-1} + u_t,\;\;\; t=1,...,T_A$$ $$y_t = \beta y_{t-1} + u_t,\;\;\;\; t=T_A+1,...,T$$ So we want to test whether $\alpha =0,\;\; t=T_A+1,...,T$ This is a test for "structural change". ...


2

In gretl, the command arima 1 3 ; y will estimate an ARMA(1,3) model on y. However, the syntax arima 1 2 3 ; y will estimate an ARIMA(1,2,3) model on y, that is, an ARMA(1,3) model on y, differenced twice. You also have ways to estimate models with seasonal filters etcetera. You'll want tpo have a look at: the help provided in the program abot the "...


2

No, cointegration occurs when two series have the same stochastic trend. In order to have a stochastic trend, they should be nonstationary. Stationary variables do not have a trend. You can perform standard inference.


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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