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So the statement is:

Adding a time trend can make an explanatory variable more significant if the dependent and independent variables have different kinds of trends, but movement in the independent variable about its trend line causes movement in the dependent variable away from its trend line.

I don't really understand the meaning. Why can variable signifcance increase?

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    $\begingroup$ Did you quote any source? In that case, please cite! $\endgroup$ Oct 12, 2023 at 15:11
  • $\begingroup$ @User1865345 I find this one chegg.com/homework-help/questions-and-answers/… which, unfortunately, provides very little other background, references, or context. $\endgroup$
    – AdamO
    Oct 12, 2023 at 18:03
  • $\begingroup$ Hm, I see @AdamO. A proper citation would have in a cogent way provided a context which might be missing in this narrow passage. It's a good precedent, in general, to always cite your source. $\endgroup$ Oct 13, 2023 at 9:25
  • $\begingroup$ The source is : Jeffrey M. Wooldridge, Introductory Econometrics a Modern Approach - 7e. Page 355. $\endgroup$ Mar 16 at 15:08

2 Answers 2

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If I could pick apart the quoted text: "different kinds of trends" is not well defined.

If we define "different kinds of trends" as $E[Y_{0,t}] \ne E[Y_{1,t}]$ for two repeatedly sampled time series, $Y_0$ and $Y_1$, at any time point $t$, then you can still test the grand hypothesis whether $\mathcal{H}_0: E[Y_0] = E[Y_1]$ using methods for independent data if the design is well balanced. That is, a plain old t-test won't hurt you. However, the choice to model the time trend can increase the power of this analysis.

Consider this simulated design

n <- 100
nt <- 4
nx <- 2

df <- expand.grid(id=1:n, t=1:nt, x=1:nx)


df$y <- with(df, 0 + 4*rnorm(nt)[t] + 1*x + rnorm(ncol(df)))


par(mfrow=c(2,1))
boxplot(y ~ x:t, data=df, at=c(0,1, 3,4, 6,7,9,10))

boxplot(y ~ x, data=df)

Here's a picture to explain:

enter image description here

Here time functions as a "blocking" factor using classic experimental design language. Even though we have defined time to be entirely random and unrelated to treatment, you can see that in groups stratified by the times 1 through 4, the observable difference between groups is far more noticeable than in the crude or aggregate scenario. This corresponds to more precise inference in ANOVA or linear regression using $p$-values:

In my simulation, the Type 3 SS test statistic for inference on the $x$ factor without adjustment for $t$ is 4 and after factor adjustment for time, it increases to 31.

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The full quote is :

In some cases, adding a time trend can make a key explanatory variable more significant. This can happen if the dependent and independent variables have different kinds of trends (say, one upward and one downward), but movement in the independent variable about its trend line causes movement in the dependent variable away from its trend line.

I did not fully follow. However I would like to share my incomplete understanding.

What is this conversation all about ? It is about spurious regression. What is spurious regression ? It results from a kind of omitted variable bias where the time trend is the omitted variable.

Now, when we omit variables correlated with one of the explanatory variables the coefficient of the explanatory variable may be biased upwards or downwards. In simple examples the bias is upwards ie on including the time trend the explanatory variable becomes less significant. The author is trying to describe a case where the explanatory variable becomes more significant on adding a time trend. That is the key point.

I think, the author is trying to say that time affects the dependent and independent variable in opposite ways. One positive and one negative.

Further, the independent variable x moves about its trend. When it moves about its trend it may cause the dependent variable y to move more or less about it's trend.

Let's us consider the case when y moves more about it's trend. I think this is what the author means when he says

movement in the independent variable about its trend line causes movement in the dependent variable away from its trend line.

Notice: in this case when x goes from below its trend to above it's trend it will induce positive correlation between y and x.

So there is positive correlation between y and x (due to movement of x around it's trend) and then there is a negative correlation between y and x due to the time trend.

When I control for the time variable, x will have a +ve coefficient.

When I ignore the time variable, the common negative correlation due to the omitted variable bias will cause the coefficient of x to be less positive.

In other words: on adding a time trend the coefficient of the explanatory variable will become more significant.

PS: I am not 100% certain of this.

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