I have panel data for 150 countries, with daily data from the start of 2018. I want to estimate the effect of the lags of one variable (x) on another variable (y), accounting for some covariates. I have selected a model I think works, but I'm not sure, and I could use some help checking that I'm not violating any key assumptions of the model.

Here is my model

(Sorry for the length.) I am using a linear regression model that includes lags of x and y as an explanatory variable, with random coefficients and random effects for each unit. Specifically:

  • The form of the equation is y_i,t = a_i + j_1 * y_i,t-1 + j_2 * y_i,t-2 + B_1 * x_i,t-1 + B_2 * x_i,t-2...
  • a_i is a unit-specific intercept / random effect (not a fixed effect, as explained below)
  • In the equation, y is regressed on the past values of y, in addition to the past values of x
  • There are also 8 other covariates / other explanatory variables with the same form as x
  • One issue is that while y, x vary day by day, some covariates vary much more slowly - e.g. year by year.
  • I run a separate regression for each unit, which includes an intercept
  • I use the weighted average of the coefficients (each unit has its own coefficients) see page 204 here
  • The coefficients are just estimated with Ordinary Least Squares.
  • I can either find a way to Wald-test the coefficients of x, or just use the variance of the coefficents of x to test for significance.

Here are my concerns about the model

Some reasons I chose this model, and some concerns I had about it:

  • The data are non-stationary, and their order of integration is 1 (first-differencing the data yields stationary data). I think this could be a problem but I'm not sure how to fix it. The Toda-Yamamoto procedure recommends only modeling with p+1 lags of x and only Wald-testing the first p lags, but I'm not sure if that applies here.
  • Because of the equation's form, the error term correlates with the first-difference of y. I think this is called Nickell bias, which I know disappears as T grows - my T is large so I think it should be okay.
  • x is lagged because it cannot have an immediate effect on y - the effect could be delayed by between 1 and 14 days (from domain knowledge)
  • The Granger Causality test is an intuitive choice for my data, and this equation has the same form as the equation used in the Granger test. Intuitively, if x has a causal effect on y, x should affect y beyond the past values of y alone.
  • I tested for serial correlation in the residuals and there isn't any, for any of the panel units.
  • 2
    $\begingroup$ Granger causality is not causality unless you make some strong assumptions. $\endgroup$
    – dimitriy
    Apr 24, 2021 at 17:18
  • $\begingroup$ I'm not using the Granger test directly - my equation just has the same form as the equation typically used in the Granger test. And in my context, there are good reasons to believe there is a causal influence if the coefficients of my model turn out to be significant. $\endgroup$ Apr 25, 2021 at 14:51


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