# Tag Info

### $I(g)$-terminology

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 ...

### Two variables: Which moves first?

"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 ...

### Why is VAR analysis all linear? Why not nonlinear?

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 ...
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### How should one determine the proper number of lags in a time series regression?

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 ...

### What does one-sided polynomial on lag operator $L$ mean?

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 ...

### Could you recommend a book or lecture notes about time series that is easy to understand?

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 ...
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### ARIMA - reason for + MA term

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 ...
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### Stationarity vs weak dependence

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 ...
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### Showing that a transformation is measure preserving

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 ...
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### Understanding the construction of stochastic processes

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. ...
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### VAR inversion - looking for a good resource

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 ...
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### HP Filter Smoothing Parameter

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 ...
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### What are accepted econometric methods to find out if a time series is I1 or not?

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 ...
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: $$\mathrm{plim}\, d^* = 2 \... 5 votes Accepted ### How to recognize correlation in spurious regression case Just regress Y on X:$$Y=b_0+b_1X+ eand 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 ... 5 votes ### Does I(1) imply a process is cointegrated with its lag? 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 ... 5 votes ### Does I(1) imply a process is cointegrated with its lag? 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://... 5 votes ### Testing for serial correlation with General Regressors My understanding is that the classical test for serial correlation is actually conditional to the validity of the strict exogeneity assumption: E[u_t|X]=0, or, as the requirement applies to any t, ... 4 votes ### Two variables: Which moves first? 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 ... 4 votes Accepted ### Demand estimation with a lagged dependent variable 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 ... 4 votes ### Why Is Cointegration Important In Practice? 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: ... 4 votes ### Looking for discussion on equilibrium vs dynamic models in econometrics 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 ... 4 votes Accepted ### Difference-in-differences with long time horizon and repeated treatments 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 ... 4 votes Accepted ### Recursive Substitution in Time Series \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} \\ &... 4 votes Accepted ### Frequency and time of a time series suddenly changed, are usual methods valid 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: ... 4 votes Accepted ### Proof that a Unit Root process is Difference Stationary It is simple to see once you factorize. In your set up there is only one unit root so the characteristic polynomial can be factorized as: \begin{align} y_t &= a_1 y_{t-1}+a_2 y_{t-2} +...+a_p y_{t-... 3 votes ### Is 10 years monthly data enough for forecasting? 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 ... 3 votes ### Missing values in economic time series 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 ... 3 votes Accepted ### How do I construct the score process of a Markov model and verify that it is a Martingale? 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 ...
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 ...