7

It's hard to tell without more context. But mostly sure it means "$Z_t$ is integrated of order $g$", i.e. $\Delta ^g Z_t=(1-d)^gZ_t$ (where $d$ is the lag operator) is stationary, in other words you have to difference $Z_t$ $g$ times to get an stationary time series. See this.


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 ...


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

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 ...


6

The book I currently use for my time series class is "Time Series Analysis and Its Applications With R Examples" by Shumway and Stoffer, 4th edition. It's served me fine, and gives lots of example codes that you can plug into R and see what happens with example data.


6

The MA terms are lagged errors (you don’t need to fetch them manually - for example in R you can use Arima function which does this for you and any program/language will have this basic function as mentioned +1 answer of @RichardHardy in +1 comment of @Dayne). For example, following Verbeek (2008) A Guide to Modern Econometrics 4th ed pp 261 ARMA(p,q) in ...


6

Weak stationarity and weak dependence are complementary conditions. A weakly stationary time series $y_t$ has an underlying statistical process which is time-invariant. This is characterised by three conditions: $$E(y_t)= \mu$$ $$E((y_t-\mu)^2)=\sigma^2 <\infty$$ $$Cov(y_t,y_{t+s})=Cov(y_t,y_{t-s})$$ Condition (1) indicates that a weakly stationary ...


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

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 ...


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

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

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 ...


5

Nerlove and Wallis (1966) result Nerlove and Wallis (1966) have discussed this issue. Their Equation (3) derives the probability limit of the Durbin-Watson statistic as: $$\mathrm{plim}\, d^* = 2 \left[ 1 - \frac{\rho \beta (\beta + \rho)}{1+\beta \rho} \right].$$ (Their notation for $\beta$ is $\alpha$ in the paper.) Nerlove and Wallis's (1966) derivation ...


5

Just regress Y on X: $$Y=b_0+b_1X+ e$$ and you will likely find some negative significant $b_1$ coefficient even though both series are just unrelated random walks. You can also see that as one series increases other one decrease so you would expect they are correlated in negative way in this case.


5

You are confusing concept of co-integration with concept of integration. If a series $y_t−y_{t−1}=u_t$ is stationary then series is integrated of order 1 not co-integrated. The term co-integration is specifically used when we have some $y_t−\beta x_{t}=u_t$ and $u_t$ is stationary, have a look at this tutorial by Reserve Bank of Australia on time series.


5

Just to add to the answers already given. I think the easiest way to see that this cannot be the case is using a CVAR formulation. For a reference see Johansen and Juselius (1990) (https://digidownload.libero.it/rocco.mosconi/JohansenJuselius1990.pdf) Consider the bivariate vector $x_t = \left\{ \begin{array}{c} y_{t}\\ y_{t-1} \end{array}\right\}$ Which we ...


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

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 ...


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 ...


4

This question is related to a post I addressed on CrossValidated. The "generalized" difference-in-differences (DiD) estimator is amenable to settings with multiple groups and multiple exposure periods. Take the following specification: $$ y_{it} = \gamma_{i} + \lambda_{t} + \delta T_{it} + \epsilon_{it}, $$ where $\gamma_{i}$ and $\lambda_{t}$ ...


4

\begin{align}y_t &= \alpha + \theta_1y_{t-1}+u_t \\ &= \alpha+\theta_1(\alpha + \theta_1y_{t-2}+u_{t-1}) + u_{t} \\ &= (1+\theta_1) \alpha + \theta_1^2y_{t-2} + \theta_1u_{t-1}+u_{t} \\ &= (1+\theta_1) \alpha + \theta_1^2(\alpha + \theta_1y_{t-3}+u_{t-3}) + \theta_1u_{t-1}+u_{t} \\ &= (1+\theta_1 + \theta_1^2) \alpha + \theta_1^3y_{t-3} ...


4

If you insist on using some standard time series model this will be problematic as standard time series models such as ARIMA require fixed frequency. There are some possible ways of dealing with this: You could try to aggregate all data on quarterly frequency. I can see that this might be somewhat more problematic due to uneven gaps on the bimonthly ...


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 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 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

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

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 ...


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