Delta method and correlated variables

Question

I have been reading about the delta method in regards to auto regressive distributed lag models. This is very new to me, so excuse any beginner mistakes.

The problem is as follows:

We have a model for gasoline consumption. $g$ is the per capita consumption, $y$ disposable income, $p$ is price, $g_{t-1}$ is lagged consumption. All the values are in logs.

$$g_t = \alpha_0 + \beta_1 p_t + \beta_2 y_t + \omega g_{t-1} + u_t$$

$\beta_i$ denote the short-run effects and $\beta_i/(1-\omega)$ denote the long-run effect. The problem is that these long-run estimates do not have standard errors calculated in most studies. I found only two papers that do: Bentzen & Engsted (2001) and Pesaran & Shin (1997). They propose to calculate the standard error using the delta method.

The problem that I see is that $y_t$ (or $p_t$) and $g_{t-1}$ are highly correlated, thus violating the delta method assumption (as far as I understand it). The correlation is quite clear since both $y_t$ and $p_t$ are significant in the regression above, so taking

$$g_{t-1} = \alpha_0 + \beta_1 p_{t-1} + \beta_2 y_{t-1} + \omega g_{t-2} + u_{t-1},$$

we know that there is correlation between $g_{t-1}$ and $p_{t-1}$ (or $y_{t-1})$, given price (or income) persistency, correlation between $g_{t-1}$ and $p_t$ or $y_t$ is surely there. I even dug a whole lot of data from Eurostat to confirm sample correlation and it was there, higher than 0.5 in absolute value.

You can also see the estimated standard error using the delta method is much larger than the ones estimated using other methods. That indicates the omitted correlation might cause the overestimation of the standard error.

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So the question is: Can I use the delta method to estimate the standard error of the non-linear transformation while knowing these variables are correlated? Or does the non-linear nature of the transformation change things?

Answer 1

Yes, you can still use the delta method with correlated variables.

Let us label your function $f(\theta)$, where $\theta = (\beta, \omega)^T$ and $f(\theta) = \beta / (1-\omega)$. The delta method is based upon the Taylor expansion:

$f(\hat{\theta}) \approx f(\theta) + (\hat{\theta} - \theta)^Tf'(\theta)$

Rearranging terms and squaring both sides results in:

$(f(\hat{\theta}) - f(\theta))^2 \approx (\hat{\theta} - \theta)^Tf'(\theta)f'(\theta)^T(\hat{\theta} - \theta)$

Taking expectations:

$\text{Var} f(\hat{\theta}) \approx \mathbb{E}(\hat{\theta} - \theta)^Tf'(\theta)f'(\theta)^T(\hat{\theta} - \theta)$

Taking derivatives of $f$ and evaluating $f'$ at $\hat{\theta}$ gives:

$f'(\hat{\theta})f'(\hat{\theta})^T = \frac{1}{(1-\hat{\omega})^2} \begin{bmatrix} 1 & \hat{\beta} / (1 - \hat{\omega}) \\ \hat{\beta} / (1 - \hat{\omega}) & \hat{\beta}^2 / (1 - \hat{\omega})^2 \end{bmatrix} $

Writing out the full expression for $\text{Var}f(\hat{\theta})$ and substituting estimates:

$\widehat{\text{Var}} f(\hat{\theta}) = \frac{1}{(1-\hat{\omega})^2}(\hat{\sigma}^2_{\beta} + 2\hat{\sigma}_{\beta \omega} \hat{\beta} / (1-\hat{\omega}) + \hat{\sigma}^2_{\omega}\hat{\beta}^2 / (1 - \hat{\omega})^2)$

You can see that positive correlation between $\beta$ and $\omega$ is going to increase the variance of the estimate of the long-run effect; it means there's a negative correlation between the estimates of $\beta$ and $1 - \omega$, the numerator and denominator of the long-run effect, so the estimated numerator and denominator tend to move in opposite directions, which naturally increases variability relative to the uncorrelated case.

Note that the delta method can fail miserably, so you might want to check its performance via simulation, e.g., by specifying all the parameters and creating many data sets with different errors, estimating the long run effect for each data set, calculating the standard deviation of the long run effect estimates, and comparing that to the delta method estimates of the standard error for the various data sets.