17

Say you have two vectors $$ \begin{align} F_{1,t} &= (y_t, y_{t-1}, y_{t-2},...) \newline F_{2,t} &= (y_t, z_t, y_{t-1}, z_{t-1},...) \end{align} $$ Then $z_t$ does not Granger cause $y_t$ if $E(y_t | F_{1,t-1}) = E(y_t | F_{2,t-1})$, i.e. $z_t$ cannot help to forecast $y_t$. So the term Granger "causality" is somewhat misleading because if a ...


15

The trade-off is between bias and power. Too few lags, you have a biased test because of the residual auto-correlation. Too many, you allow for potentially spurious rejections of the null - some random correlation might make it look like $X$ helps predict $Y$. Whether or not that's a practical concern depends on your data, my guess would be to lean higher, ...


14

Granger causality is essentially usefulness for forecasting: X is said to Granger-cause Y if Y can be better predicted using the histories of both X and Y than it can by using the history of Y alone. GC has very little to do with causality in Pearl's counterfactual sense, which involves comparisons of different states of the world that could have occurred. ...


12

Pearl provides a calculus for reasoning about causality, Granger provides a method for discovering potential causal relations. I will elaborate: Pearl's work is based on what he has termed "Structural Causal Models", which is a triple M = (U, V, F). In this model U is the collection of the exogenous (background, or driving) unobserved variables, V is the ...


11

The help page for the grangertest function is pretty clear, it should be of major help. Model 1 is the unrestricted model that includes the Granger-causal terms. Model 2 is the restricted model where the Granger-causal terms are omitted. The test is a Wald test that assesses whether using the restricted Model 2 in place of Model 1 makes statistical sense (...


9

Here's what I understand the situation is. You have one model, which you call your simulation, that you are confident generates a set of data that accurately represents what will actually happen in the epidemic. For some reason (presumably because it's expensive or slow to build and run, or there's theoretical interest in a simple equation that generates ...


8

Introduction This test in your question seems rather heavy handed. It is conducting pairwise bivariate Granger causality testing over all pairs in the data set. I'll choose two to examine. require(lmtest) ts(datSel$cpi)->cpi ts(datSel$lohn)->wages #i presume Note that in your test at lag order 4 we get that both wages Granger cause cpi and cpi ...


8

Granger causality tells you that variable $X$ provides helpful information about future values of $Y$ <...> The devil is in the details. Granger causality considers the incremental benefits on forecasting $Y$ due to using the history of $X$ extra to using the history of $Y$ alone. That is, the benchmark forecast is based on past values of $Y$ alone ...


7

If you mean econometric theory, I would not be surprised to learn that most of the relevant and needed theoretical results related to the notion of Granger causality have already been derived. After all, the notion of Granger causality is nearly 50 years old. Hence, I do not expect to see much research on the theoretical aspects of Granger causality any more....


6

Follow this procedure (Engle-Granger Test for Cointegration): 1) Test to see if your series are stationary using adfuller test (stock prices and GDP levels are usually not) 2) If they are not, difference them and see if the differenced series are now stationary (they usually are). 3) If they are, your ORIGINAL series are said to be each integrated (I did ...


6

A non-linear Granger causality test was implemented by Diks and Panchenko (2006). The code can be found here and it is implemented in C. The test work as follows: Suppose we want to infer about the causality between two variables $X$ and $Y$ using $q$ and $p$ lags of those variables, respectively. Consider the vectors $X_t^q = (X_{t-q+1}, \cdots, X_t)$ and $...


6

Some types of nonstationarity are allowed, as long as we can build a model and a testing procedure that account for the specific type of nonstationarity. See Dave Giles' famous blog post "Testing for Granger causality" for the case of unit-root nonstationarity. But obviously, not all types of nonstationarity can be allowed for. If the time series is too ...


6

[W]hat is this Granger test for and how to interpret it? Basically, Granger causality $x \xrightarrow{Granger} y$ exists when using lags of $x$ next to the lags of $y$ for forecasting $y$ delivers better forecast accuracy than using only the lags of $y$ (without the lags of $x$). You can find definitions and details in Wikipedia and in free textbooks and ...


4

Consider the following examples: Your time series are the gross domestic products (GDP) of France and Germany. As the two country’s economics are strongly interacting, a strong French economy is likely to give rise to an improvement in the German economy and vice versa. Thus, knowing the GDP for France allows you to better predict the German GDP and vice ...


4

You can certainly calculate the correlation between two time series. That's a short answer. When, as true here and as true often, there is a marked trend in both cases, the correlation is likely to be extremely high. In general, it's not especially helpful. It's not as if there was serious doubt that there would be an apparent association; that's easily ...


4

If you have two time series, then the univariate unit-root test results coincide nicely with the cointegration test results. Both indicate that the series are stationary. Moreoever, this is as expected; financial returns are normally considered not to have unit roots. However, you could think about cointegration between the stock price and the level of the ...


4

Interpretability is another issue. While you are of course right that structural responses are generally of more interest, even an orthogonal impulse response generally is more useful than the estimated VAR coefficients simply because it is easier to see the dynamic response of the variables to a shock in one variable. Here is an example from the vars ...


4

I was looking for the answer to this same question and I found it on the book Introduction to Modern Time Series Analysis (second edition) by Gebhard Kirchgassner, Jurgen Wolters and Uwe Hassler on page 97. Granger Causality: x granger causes y if a model that uses current and past values of x and current and past values of y to predict future values of y ...


3

Yes. In the specific case of VAR models, imagine we want to test the restriction that $x$ does not Granger-cause $y$. We have the regression (in $y$): $$y_t=\alpha + \sum_{l=1}^p \delta_ly_{t-l}+\sum_{l=1}^q \gamma_lx_{t-l}+\epsilon_t$$ The corresponding null to our Granger causality test is $H_0:\gamma_l=0,\,\,l=1,\dots,p$. This is a Wald test, and under ...


3

I fully agree with Andy, and I was actually thinking of writing something similar, but then I started to wonder myself about this topic. I think we all agree that Granger causality itself really has not much to do with causality as understood in the potential outcomes framework, simply because Granger causality is more about time precedence than anything ...


3

When I first used the causality function of the vars package I had the same doubt. Here is what I thought. Imagine a trivariate VAR(1) model: $$ Y_t = a_0 + a_1 Y_{t-1} + a_2 X_{t-1} + a_3 Z_{t-1} + \epsilon_{y,t} \\ X_t = b_0 + b_1 Y_{t-1} + b_2 X_{t-1} + b_3 Z_{t-1} + \epsilon_{x,t} \\ Z_t = c_0 + c_1 Y_{t-1} + c_2 X_{t-1} + c_3 Z_{t-1} + \epsilon_{z,t} $...


3

Yes, absolutely. CausalImpact constructs a counterfactual to the observed post-intervention outcomes using a combination of all the control time series you enter. So in practice you almost always want at least a few control time series. Keep in mind that the model assumes all of these to be unaffected by the treatment. The package documentation has more ...


3

Consider the definition of cointegration: There exists a linear combination of several time series that is stationary despite the fact that each of the individual series is nonstationary. Hence, per definition, cointegration can only occur if the individual series are nonstationary. To check for a relationship among stationary time series, you can, for ...


3

Positive semidefinitness of the difference of two matrices is the concept that is analogous to the difference being non-negative for unidimensional entities. In particular, a positive semidifinitness matrix has all its diagonal elements non-negative. Diagonal elements here represent same-period forecast MSE's so positive definitness says that the forecast ...


3

From the perspective of testing for Granger causality, your setup is exactly the same as a VAR(1) setup if equation-by-equation OLS is used for estimating the restricted VAR. The coefficient estimates and their standard errors in your individual equations will be obtained in exactly the same way as in the VAR(1) model and will have the same values, both in ...


3

Why would you use Granger causality rather than a regular regression and see if the coefficient of the variable in the preceding time frame is significant? You actually do that in Granger causality testing, but the "regular regression" must include the autoregressive (=own) lags per definition of Granger causality. Granger causality considers whether the ...


3

"Granger causality" has nothing to do with causality. It is just that the late Sir Clive Granger was a master of marketing his tremendously important work by tremendously catchy names. We should remember that prof. Granger was heavy into forecasting. And in forecasting, if I can forecast your birthday through your (or my) mom, I am perfectly fine. Meaning,...


3

Scatter plots between the original series can often be useful but of more importance is scatter plots conditional on data conditioned for temporal activities. Often one needs to allow for hourly or daily effects (be they stochastic or deterministic ) and latent level shifts/time trends in order to tease out (identify) useful predictor structure for user ...


3

Granger causality can be applied to binary data by using the appropriate univariate distributions. For instance, in the two-series case you could assume the following structure. Let $\mathbf{y}_t = (x_t, y_t)$ be the vector of binary data at time $t \in \mathbb{N}$. Assume an autoregressive structure (like VAR) as follows: each component is independently ...


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