# Residual plots: why plot versus fitted values, not observed $Y$ values?

In the context of OLS regression I understand that a residual plot (vs fitted values) is conventionally viewed to test for constant variance and assess model specification. Why are the residuals plotted against the fits, and not the $$Y$$ values? How is the information different from these two plots?

I am working on a model that produced the following residual plots:

So the plot vs the fitted values looks good at quick glance, but the second plot against the $$Y$$ value has a pattern. I'm wondering why such a pronounced pattern wouldn't also manifest in the residual vs fit plot....

I'm not looking for help in diagnosing issues with the model, but just trying to understand the differences (generally) between (1) residual vs fit plot & (2) residual vs $$Y$$ plot.

For what it's worth, I'm sure the error pattern in the second chart is due to omitted variable(s) which influence the DV. I'm currently working on obtaining that data, which I expect will help the overall fit and specification. I am working with real estate data: DV=Sales Price. IVs: Sq.ft of house, # garage spaces, year built, year built$$^2$$.

• I've taken the liberty of tweaking the title to match your intent a little more closely. Even among economists (you may be one) "IV" has another meaning of instrumental variable, although there is no ambiguity in this case. For better communication across several statistical sciences, some of us discourage locally used abbreviations such as DV (which for some people still means Deo volente) and IV in favour of evocative terms such as response or outcome on the one hand and predictor or covariate on the other. I know this is a detail in your question, but it has been well answered. Jun 5, 2015 at 7:41

By construction the error term in an OLS model is uncorrelated with the observed values of the X covariates. This will always be true for the observed data even if the model is yielding biased estimates that do not reflect the true values of a parameter because an assumption of the model is violated (like an omitted variable problem or a problem with reverse causality). The predicted values are entirely a function of these covariates so they are also uncorrelated with the error term. Thus, when you plot residuals against predicted values they should always look random because they are indeed uncorrelated by construction of the estimator. In contrast, it's entirely possible (and indeed probable) for a model's error term to be correlated with Y in practice. For example, with a dichotomous X variable the further the true Y is from either  E(Y | X = 1) or E(Y | X = 0) then the larger the residual will be. Here is the same intuition with simulated data in R where we know the model is unbiased because we control the data generating process:

set.seed(21391209)

trueSd <- 10
trueA <- 5
trueB <- as.matrix(c(3, 5, -1, 0))
sampleSize <- 100

# create independent x-values
x1 <- rnorm(n=sampleSize, mean = 0, sd = 4)
x2 <-  rnorm(n=sampleSize, mean = 5, sd = 10)
x3 <- 3 + x1 * 4 + x2 * 2 + rnorm(n=sampleSize, mean = 0, sd = 10)
x4 <- -50 + x1 * 7 + x2 * .5 + x3 * 2  +
rnorm(n=sampleSize, mean = 0, sd = 20)
X = as.matrix(cbind(x1, x2, x3, x4))

# create dependent values according to a + bx + N(0,sd)
Y <-  trueA +  X %*%  trueB  +
rnorm(n=sampleSize, mean=0, sd=trueSd)

df = as.data.frame(cbind(Y, X))
colnames(df) <- c("y", "x1", "x2", "x3", "x4")
ols = lm(y ~ x1 + x2 + x3 + x4, data = df)
y_hat = predict(ols, df)
error = Y - y_hat
cor(y_hat, error) #Zero
cor(Y, error) #Not Zero


We get the same result of zero correlation with a biased model, for example if we omit x1.

ols2 = lm(y ~ x2 + x3 + x4, data = df)
y_hat2 = predict(ols2, df)
error2 = Y - y_hat2
cor(y_hat2, error2) #Still zero
cor(Y, error2) #Not Zero

• Helpful, but the first sentence could be rewritten for clarity. "Construction" produces the residuals; the error term is considered to be out there and in existence prior to calculation. Similarly, I would say that it's the estimates that are constructed, not the estimator, which is the method used to construct them. Jun 5, 2015 at 7:33
• But then why do we even look at the residual chart (vs fits)? What diagnostic purpose does that plot have? I'm new to the site. Do I have to tag Michael or does he get this comment automatically? My comment would also apply to @Glen_b answer below. Both answers help my understanding. Thanks.
– Mac
Jun 5, 2015 at 12:02
• ... because they may reveal other structure. The lack of correlation between residual and fit doesn't mean that other things can't be happening too. If you believe your model is perfect then you won't believe that to be possible....In practice you do need to check for other kinds of structure. Jun 5, 2015 at 12:07
• @Mac, I'll be honest and say that I never look at these plots. If you're trying to make a causal inference then you should think through omitted variable problems and reverse causality problems conceptually. Either problem could occur and you would not be able to eye ball it from these plots as they are problems of observational equivalence. If all you care about is prediction then you should think through and test out-of-sample how well your model's predictions perform out-of-sample (otherwise it's not a prediction). Jun 5, 2015 at 16:14
• I disagree strongly; my experience is quite otherwise. I have used such plots to spot (e.g.) nonlinearity, heterosedasticity, granular and group structure not evident in output. It is evident from texts and papers that many researchers have found them useful for far more than just spotting outliers. The evidence of what does and does not work is vital in assessing models. Jun 5, 2015 at 16:37

Two facts which I assume you're happy with me just stating:

i. $$y_i = \hat{y}_i+\hat{e}_i$$

ii. $$\text{Cov}(\hat{y}_i,\hat{e}_i)=0$$

Then:

$$\text{Cov}(y_i,\hat{e}_i)=\text{Cov}(\hat{y}_i+\hat{e}_i,\hat{e}_i)$$

$$\qquad=\text{Cov}(\hat{y}_i,\hat{e}_i) +\text{Cov}(\hat{e}_i,\hat{e}_i)$$

$$\qquad=0 +\sigma^2_e$$

$$\qquad=\sigma^2_e$$

So while the fitted value isn't correlated with the residual, the observation is.

In effect, this is because both the observation and the residual are related to the error term.

This usually makes it somewhat harder to use the plot of residuals vs observations for diagnostic purposes; the addition of a linear relationship (and dependence) to the deviation from a linear relationship tends to partially disguise the pattern in the second thing (it's harder to 'see' what's going on).