# Diagnostics part 2: error covariance assumptions

10/21/2020

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## Outline

• The following topics will be covered in this lecture:
• A review of the residuals and their covariance
• Visual diagnostics of non-constant variances
• The F-test for non-equal variances
• Correlation between cases
• The Durbin-Watson test for time correlation

## Diagnostics

• We are considering 4 categories of potential issues with the model:
1. Issues with the Gaussian error assumption: our hypothesis testing studied thus far relies on the Gaussian error assumption.
2. Issues with the form of the hypothetical covariance: we have assumed that $\mathrm{cov}\left(\mathbf{Y}\right) = \sigma^2\mathbf{I}$ but for many cases this will not be true.
3. Issues with unusual observations: some of the observations may not look like others, and they might change the choice and the fit of the model.
4. Issues with the systematic part of the model: we have assumed that there is an actual signal in the data of the form \begin{align} \mathbb{E}[\mathbf{Y}] = \mathbf{X} \boldsymbol{\beta}, \end{align} which may not be valid.

## Checking error covariance assumptions

• We will now begin our discussion on diagnosing issues with our error covariance assumptions.

• If we wish to check the assumptions on the error or variation in the signal $$\boldsymbol{\epsilon}$$, we need to consider, $$\boldsymbol{\epsilon}$$ itself is not observable.

• Q: What proxy could we consider for the error?

• A: The residuals are related to the error functionally, but have slightly different properties.
• Recall, the definition of $$\hat{\mathbf{Y}}$$

\begin{align} \hat{\mathbf{Y}} \triangleq& \mathbf{X}\left(\mathbf{X}^\mathrm{T} \mathbf{X}\right)^{-1} \mathbf{X}^\mathrm{T} \mathbf{Y} \\ & =\mathbf{H} \mathbf{Y} \end{align}

• Therefore, if we compute the residuals,

\begin{align} \hat{\boldsymbol{\epsilon}} & = \mathbf{Y} - \hat{\mathbf{Y}} \\ & =\left(\mathbf{I} - \mathbf{H}\right)\mathbf{Y} \\ & =\left(\mathbf{I} - \mathbf{H}\right)\mathbf{X} \boldsymbol{\beta} + \left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon} \end{align}

### Checking error covariance assumptions

• From the last slide

\begin{align} \hat{\boldsymbol{\epsilon}} & =\left(\mathbf{I} - \mathbf{H}\right)\mathbf{X} \boldsymbol{\beta} + \left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon} \end{align}

• Q: recalling the definition of $$\mathbf{H} = \mathbf{X}\left(\mathbf{X}^\mathrm{T} \mathbf{X}\right)^{-1} \mathbf{X}^\mathrm{T}$$, what does $$\mathbf{H}\mathbf{X}$$ equal to?

• A: $$\mathbf{H}$$ is the projection operator onto the span of the design matrix, and thus $$\mathbf{H}\mathbf{X} = \mathbf{X}$$ by construction.

• Q: given the above, what does $$\left(\mathbf{I} - \mathbf{H}\right)\mathbf{X}$$ equal to?

• A: the above must equal zero, as $$\mathbf{I}\mathbf{X} = \mathbf{H}\mathbf{X} = \mathbf{X}$$.

### Checking error covariance assumptions

• From the previous two exercises, we can deduce,

\begin{align} \hat{\boldsymbol{\epsilon}} = \left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon} \end{align}

• We take the assumption that $$\boldsymbol{\epsilon}\sim N(0, \mathbf{I} \sigma^2)$$,

• Q: given the above assumption, what is the mean of $$\hat{\boldsymbol{\epsilon}}$$?
• A:

\begin{align} \mathbb{E}[\hat{\boldsymbol{\epsilon}}] &= \mathbb{E}\left[\left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon}\right] \\ &=\left(\mathbf{I} - \mathbf{H}\right)\mathbb{E}\left[\boldsymbol{\epsilon}\right]\\ &= 0 \end{align}

• Q: given the above assumption, what is the covariance of $$\hat{\boldsymbol{\epsilon}}$$?
• A:

\begin{align} \mathbb{E}\left[\left(\hat{\boldsymbol{\epsilon}}\right) \left(\hat{\boldsymbol{\epsilon}}\right)^\mathrm{T}\right] &= \mathbb{E}\left[\left(\left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon}\right)\left(\left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon}\right)^\mathrm{T}\right] \\ &=\mathbb{E}\left[\left(\mathbf{I} - \mathbf{H}\right)\boldsymbol{\epsilon}\boldsymbol{\epsilon}^\mathrm{T}\left(\mathbf{I} - \mathbf{H}\right)^\mathrm{T}\right]\\ &=\left(\mathbf{I} - \mathbf{H}\right)\mathbb{E}\left[\boldsymbol{\epsilon}\boldsymbol{\epsilon}^\mathrm{T}\right]\left(\mathbf{I} - \mathbf{H}\right)^\mathrm{T}\\ & =\left(\mathbf{I} - \mathbf{H}\right)\left[\sigma^2 \mathbf{I}\right]\left(\mathbf{I} - \mathbf{H}\right)\\ &=\sigma^2\left(\mathbf{I} - \mathbf{H}\right) \end{align}

### Checking error covariance assumptions

• Q: Why is the last slide relevant? Particularly, why should we be concerned with the covariance of the residuals?

• A: If the assumptions hold for $$\boldsymbol{\epsilon}$$, then we can compare the coviance of the residuals to their theoretical value $$\sigma^2\left(\mathbf{I} - \mathbf{H}\right)$$ for consistency.
• Occasionally, we might actually have prior knowledge about the value of $$\sigma^2$$ which we can evaluate directly.
• Otherwise, we have an unbiased estimator for $$\sigma^2$$ in terms of $\hat{\sigma}^2 = \frac{RSS}{n-p}$ which we can compare with the observed covariance of the residuals.
• Note, even though the errors are assumed to be independent and of equal variance $$\sigma^2$$, the same doesn't hold for the residuals.

• In particular, the operator $$\mathbf{I} - \mathbf{H}$$ is not generally diagonal (such that the residuals have correlation); nor are the diagonal values equal (so that the variances don't all match).
• Nonetheless, we use the residuals to underrstand how the true errors are behaving, which are unobservable.

## Visual diagnostics Courtesy of: Faraway, J. Linear Models with R. 2nd Edition

• To the left is a plot where:
1. The horizontal (x-axis) represents the fitted value $$\hat{\mathbf{Y}}$$, as in some model $$\hat{\mathbf{Y}}=\mathbf{X}\hat{\boldsymbol{\beta}}$$.
2. The vertical axis (y-axis) represents the corresponding residual $$\hat{\boldsymbol{\epsilon}}$$ value, equal to $$\mathbf{Y} - \hat{\mathbf{Y}}$$.
• Suppose that the variances of the error $$\boldsymbol{\epsilon}$$ are not fixed, i.e.,
• suppose that some observations have more variation and some observations have less variation around the signal, $$\mathbf{X}\boldsymbol{\beta}$$.
• In this case, there is dependence of the variation (or error $$\boldsymbol{\epsilon}$$) on the observation.
• To the left, this is actuall a well behaved situation. In particular, the residuals show variation that doesn’t seem to depend on the value of the fitted value/ observation.
• The mean of the residuals appears to be zero as well, because there isn’t a clear preference for positive or negative residuals
• The situation to the left is denoted homoscedasticity.
• Later, we will develop tests to determine if the variance is “close-enough-to-constant”, so that we are satisfied.
• On the other hand, if there are non-constant variances (as discussed above), this will often show up as a pattern in the plot to the left.

### Visual diagnostics Courtesy of: Faraway, J. Linear Models with R. 2nd Edition

• The non-constant variance of $$\hat{\boldsymbol{\epsilon}}$$ to the left is known as heteroscedasticity.
• In this case, there is a clear dependence on the variation of $$\hat{\boldsymbol{\epsilon}}$$ on the fitted value/ the observation.
• Breaking the constant variance assumption, the Gauss-Markov theorem no longer applies
• Heteroscedasticity does not, in itself, cause ordinary least squares coefficient estimates to be biased;
• however, the theoretical estimates of the variance of the residuals (and therefore the standard errors) will become biased.

### Visual diagnostics Courtesy of: Faraway, J. Linear Models with R. 2nd Edition

• The bias in the standard errors complicates our ability to accurate quantify the uncertainty, and therefore to accurately:
1. make hypothesis tests on the parameters for significance;
2. provide confidence intervals for the parameters
3. provide accurate prediction intervals and confidence intervals for the mean response;
4. provide explanatory power in the relationship between the response and the explanatory variables.

### Visual diagnostics Courtesy of: Faraway, J. Linear Models with R. 2nd Edition

• Using the same plot as before, we can likewise determine if there is actually nonlinearity in the residuals.
• The plot at the left exhibits a nonlinear dependence of the residual on the fitted/ observed values
• The same technique can also be used replacing the fitted values $$\hat{\mathbf{Y}}$$ in the horizontal axis with any other variable in the model $$\mathbf{X}_i$$, to determine dependence of the residual on the explanatory variables
• Additionally, we may consider how the residual varies across variables $$\mathbf{X}_i$$ that we have data for, but have not included as explanatory variables on the response.
• If there is a dependence structure for the residuals on the variable that was left out of the model, it suggests that we should consider its impact on the response and/or if it is a variable that is tightly correlated with our existing model variables.

## An example of non-constant variance

• Recall the “savings” dataset, listing the average savings rate, age demographics and per capita incomes of fifty countries averaged over 1960 -1970.
library("faraway")
lmod_savings <- lm(sr ~ pop15+pop75+dpi+ddpi,savings)
sumary(lmod_savings)

               Estimate  Std. Error t value  Pr(>|t|)
(Intercept) 28.56608654  7.35451611  3.8842 0.0003338
pop15       -0.46119315  0.14464222 -3.1885 0.0026030
pop75       -1.69149768  1.08359893 -1.5610 0.1255298
dpi         -0.00033690  0.00093111 -0.3618 0.7191732
ddpi         0.40969493  0.19619713  2.0882 0.0424711

n = 50, p = 5, Residual SE = 3.80267, R-Squared = 0.34

• We saw in the last lecture that the residuals do not diverge strongly from the Gaussian assumption, but some of the predictors themselves appear to have skew or multimodal distributions.

• We will be interested now in examining our error covariance assumptions.

### Savings data example

• With this model, we plot the residuals versus the fitted values:
par(mai=c(1.5,1.5,.5,.5), mgp=c(3,0,0))
plot(fitted(lmod_savings),residuals(lmod_savings),xlab="Fitted",ylab="Residuals", cex=3, cex.lab=3, cex.axis=3)
abline(h=0) • From the initial inspection, we do not notice anything particularly structured about the residuals – indeed they are roughly symmetric around zero and in the fitted value.

### Half normal distributions Courtesy of Nagelum CC BY-SA 4.0 via Wikimedia Commons
• Therefore, to investigate the constant variance assumption more closely, we will “zoom” in on the residuals.
• In particular, we will consider the plot of the square root of the absolute residuals, i.e., \begin{align} \sqrt{\vert\hat{\boldsymbol{\epsilon}}\vert} \end{align}
• Under the assumption $$\boldsymbol{\epsilon}\sim N(0, \mathbf{I}\sigma^2)$$, the values of $$\vert\hat{\boldsymbol{\epsilon}}\vert$$ are distribted according to the half normal distribution
• the relationship between the normal distribution and the corresponding half-normal distribution is illustrated on the left.
• Having excluded the nonlinearity expressed in other plots, we will focus on the absolute values to increase the resolution of the residuals as a response of the fitted values.
• However, the half normal distribution is very skewed, so we make a change of scale with the square root of the absolute values in order to keep it well behaved.
• ### Savings data example

• In the case below, we plot $$\sqrt{\vert\hat{\boldsymbol{\epsilon}}\vert}$$ versus the fitted values
par(mai=c(1.5,1.5,.5,.5), mgp=c(3,0,0))
plot(fitted(lmod_savings),sqrt(abs(residuals(lmod_savings))), xlab="Fitted",ylab=expression(sqrt(hat(epsilon))), cex=3, cex.lab=3, cex.axis=1.5)