R Packages/Data for This Session

We’ll be using {tidyverse} for all things workflow and {stevedata} for toy data. You may also want to install {stevemisc} from Github, though we’ll load that cor2data() function here so you don’t have to do this.

library(tidyverse)

## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.0 ──

## ✓ ggplot2 3.3.3     ✓ purrr   0.3.4
## ✓ tibble  3.0.5     ✓ dplyr   1.0.3
## ✓ tidyr   1.1.2     ✓ stringr 1.4.0
## ✓ readr   1.4.0     ✓ forcats 0.5.0

## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(stevedata)


# You can either install and load stevemisc, or call this function randomly. Your pick.
cor2data <- function(cor, n, seed){
  # number of observations to simulate
  nobs = n
  
  # Cholesky decomposition
  U = t(chol(cor))
  nvars = dim(U)[1]
  
  if(missing(seed)) {
    
  } else {
    set.seed(seed)
  }
  
  # Random variables that follow the correlation matrix
  rdata = matrix(rnorm(nvars*nobs,0,1), nrow=nvars, ncol=nobs)
  X = U %*% rdata
  # Transpose, convert to data
  # require(tidyverse)
  Data = t(X)
  Data = as.data.frame(Data)
  return(Data)
}

OLS with Fake Data

Let’s create a few data sets from the cor2data() function. This should look very familiar to what we did the previous week with measurement error. We’ll keep it simple here and have just the one predictor (x1) uncontaminated by any endogeneity concerns.

vars = c("x1",  "e")
Cor <- matrix(cbind(1, 0.001,
                    0.001, 1), nrow=2)
rownames(Cor) <- colnames(Cor) <- vars

A <- cor2data(Cor, 1000, 8675309) %>% as_tibble() # Jenny I got your number...

We’ll set up a simple situation where the outcome y is a function of this formula. The expected value of y is 1 when x1 is 0. The errors are kind of “skinny” too, at least relative to the effect of x1.

A$y <- with(A, 1 + x1 + .5*e)

Okay then, let’s do a regression to show what we just did.

summary(M1 <- lm(y ~ x1, data=A))

## 
## Call:
## lm(formula = y ~ x1, data = A)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.9199 -0.3400  0.0034  0.3278  1.7039 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.00204    0.01620   61.86   <2e-16 ***
## x1           0.99107    0.01582   62.66   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5122 on 998 degrees of freedom
## Multiple R-squared:  0.7973, Adjusted R-squared:  0.7971 
## F-statistic:  3926 on 1 and 998 DF,  p-value: < 2.2e-16

^ Nailed it.

Btw, if you just wanted to do a simple lm with one y on one x, ggplot could do this for you.

A %>%
  ggplot(.,aes(x1,y)) +
  geom_point() +
  geom_smooth(method ="lm")

## `geom_smooth()` using formula 'y ~ x'

One of the properties of the SRF with one x is the regression line will travel through the mean of y and x. Observe:

a_meany = mean(A$y)
a_meanx1 = mean(A$x1)

A %>%
  ggplot(.,aes(x1,y)) +
  geom_point() +
  geom_smooth(method ="lm") +
  geom_vline(xintercept = a_meanx1) +
  geom_hline(yintercept = a_meany)

## `geom_smooth()` using formula 'y ~ x'

One of the assumptions is the mean of residuals will be zero and x is uncorrelated with the residuals.

A %>%
  mutate(resid = resid(M1)) %>%
  summarize(corx1 = cor(resid, x1),
            meanresid = mean(resid))

## # A tibble: 1 x 2
##      corx1 meanresid
##      <dbl>     <dbl>
## 1 4.78e-17  8.38e-18

The Basic Diagnostic Checks for Beginners

Always, always, always do some diagnostics on your OLS model. This will, in practice, mean looking at your residuals and extracting your fitted variables. Let’s do that:

A %>%
  mutate(resid = resid(M1),
         fitted = fitted(M1)) -> A

Compare the fitted versus the residual. Fitted vs. residual plots should more or less look random. The line through it should be flat/straight through 0 on the y-intercept

A %>%
  ggplot(.,aes(fitted, resid)) +
  geom_point() +
  geom_smooth(method="lm") +
  geom_hline(yintercept = 0, linetype="dashed")

## `geom_smooth()` using formula 'y ~ x'

The second plot is a normal Q-Q plot of the residuals. If the the line is straight, it suggests the errors are normally distributed

A %>%
  ggplot(.,aes(sample=resid)) + geom_qq()

The third plot is a scale-location plot. Basically: plot the fitted values against the square root of the absolute values of the residuals. This too should effectively be random

A %>% 
  mutate(absqresid = sqrt(abs(resid))) %>%
  ggplot(.,aes(fitted, absqresid)) +
  geom_point() + geom_smooth(method = "lm")

## `geom_smooth()` using formula 'y ~ x'

The fourth plot is a residuals vs. leverage plot. This will help you find any influential cases in your data. These residuals vs. leverage plots are typically communicated with standardized residuals

Some definitions:

outlier: an observation with a large residual
leverage point: an observation with an x value far from the mean.
influence observation: an observation that changes the slope of the line

One way around this is to compare the regression line with and without those observations.

A %>%
  mutate(stdresid = rstandard(M1), # standardized residuals, via handy built-in function
         lev = hatvalues(M1)) %>% # leverages. Trust me, just do it this way. matrix algebra is a pain.
  ggplot(.,aes(lev, stdresid)) +
  geom_point() +
  geom_smooth(method ="loess")

## `geom_smooth()` using formula 'y ~ x'

One thing happening underneath the proverbial hood of these diagnostic plots is a Cook’s distance score. Cook’s distance is a measure used to calculate influence in OLS. Higher values of Cook’s D = more influence. So what defines “high” and “influencer?” Some studies recommend a threshold of 4/n, or 4/(n-k-1). Alternatively: check if it’s above 1 if your data are sufficiently large. Values above that are problematic. Even then, so many rules of thumb. My hot #take: this is a “use your head” problem, following a graph of Cook’s D scores. And you’ll need to eyeball this, as I’ve already done:

tibble(obs = 1:1000,
       cooksd = cooks.distance(M1)) %>%
  bind_cols(A, .) -> A

A %>%
  ggplot(.,aes(obs, cooksd)) + geom_bar(stat="identity") +
  geom_hline(yintercept = .01, linetype="dashed") +
  geom_text(data=subset(A, cooksd>.01), aes(y=cooksd, label=obs), size=4)

Traditional logic (and this is still valid): if you have problematic observations/influencers, punt them out, re-run the model, and compare the results. My hot #take: if you do that, the Cook’s D scores for the new regression will change as well. Cook’s D is kind of relative. Basically, take a look, find influencers/outliers, but don’t try to wed yourself to Cook’s D as a diagnostic. Base R has a lot of plots that could do this manually for you. It won’t be as pretty, and you shouldn’t present them “as is”, but they’re a little convenient. FYI: I hate built-in functions that make me hit return to see the next plot. It’s why I did this the ggplot way.

plot(M1)

This will return the four biggies that I described above, but you could grab an individual one, if you’d like. See:

plot(M1, which=1)

Want the Cook’s D histogram? That’s no. 4.

plot(M1, which=4)

There is one other too:

plot(M1, which=6)

It plots the leverage points against Cook’s D. Never have been a big fan of this plot.

A Replication of Newhouse (1977)

Let’s replicate Newhouse (1977) now. I like this regression for this purpose not because it’s good, but because you won’t find many political science articles where the full damn data for the regression is in the article itself. That’s pretty cool and the topic is still accessible even if the analysis is clearly dated.

The data are in {stevedata} as follows.

Newhouse77

## # A tibble: 13 x 5
##    country         year gdppc medsharegdp medexppc
##    <chr>          <dbl> <dbl>       <dbl>    <dbl>
##  1 Australia       1972  3769        5.8       219
##  2 Austria         1972  2747        5.46      150
##  3 Canada          1971  4317        7.02      303
##  4 Finland         1972  2869        5.31      152
##  5 France          1970  2851        6.4       182
##  6 GFR             1968  2246        6.07      136
##  7 Greece          1972  1374        2.51       34
##  8 Italy           1972  2164        6.12      132
##  9 Netherlands     1972  3437        6.78      233
## 10 Norway          1972  3889        5.18      201
## 11 Sweden          1971  4431        7.38      327
## 12 United Kingdom  1972  2742        4.45      122
## 13 United States   1972  5551        6.51      361

?Newhouse77 # for background info

Okie dokie. Let’s reproduce Table 2 now. Here, Newhouse is regressing medical care expenditures per capita as a function of GDP per capita. In his analysis, he does go-go-gadget OLS on the full data but then does another analysis that excludes Greece. In his analysis, Greece is apparently an outlier than can monkey with inferences.

summary(M1 <- lm(medexppc ~ gdppc , data=Newhouse77))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = Newhouse77)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -44.850 -15.866  -5.825  22.133  38.424 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -60.722513  23.598989  -2.573   0.0259 *  
## gdppc         0.078831   0.006873  11.470 1.85e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 26.67 on 11 degrees of freedom
## Multiple R-squared:  0.9228, Adjusted R-squared:  0.9158 
## F-statistic: 131.6 on 1 and 11 DF,  p-value: 1.849e-07

summary(M2 <- lm(medexppc ~ gdppc, data=subset(Newhouse77, country !="Greece")))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = subset(Newhouse77, country != 
##     "Greece"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -44.814 -16.364   3.409  18.844  39.803 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -51.118202  29.167401  -1.753     0.11    
## gdppc         0.076352   0.008212   9.297 3.09e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.49 on 10 degrees of freedom
## Multiple R-squared:  0.8963, Adjusted R-squared:  0.8859 
## F-statistic: 86.44 on 1 and 10 DF,  p-value: 3.086e-06

Basic takeaway maps to what Newhouse reports. Higher GDP per capita > higher per capita medical care expenditures. This holds with or without Greece.

Question, for my own curiosity: is Greece that much an outlier? It doesn’t really look like it. It’s an outlier in that it scores low on both the DV and the IV and not necessarily that it’s removed from what would be expected, given the data. If Greece is a problem observation, shouldn’t the U.S. be a problem observation as well? Greece scores as low as the U.S. scores high on both the IV and DV. Let’s plot the data (which you should always do for simple cases like this). Greece is pretty easy to identify here. But, again, it’s very much in line with what would be expected.

Newhouse77 %>% 
  ggplot(.,aes(gdppc, medexppc)) + 
  geom_point() + geom_smooth(method = "lm")

## `geom_smooth()` using formula 'y ~ x'

So, to answer this question, what would the Cook’s D say?

Newhouse77 %>%
  mutate(cooksd = cooks.distance(M1))

## # A tibble: 13 x 6
##    country         year gdppc medsharegdp medexppc  cooksd
##    <chr>          <dbl> <dbl>       <dbl>    <dbl>   <dbl>
##  1 Australia       1972  3769        5.8       219 0.0244 
##  2 Austria         1972  2747        5.46      150 0.00275
##  3 Canada          1971  4317        7.02      303 0.0807 
##  4 Finland         1972  2869        5.31      152 0.0133 
##  5 France          1970  2851        6.4       182 0.0240 
##  6 GFR             1968  2246        6.07      136 0.0541 
##  7 Greece          1972  1374        2.51       34 0.0862 
##  8 Italy           1972  2164        6.12      132 0.0759 
##  9 Netherlands     1972  3437        6.78      233 0.0340 
## 10 Norway          1972  3889        5.18      201 0.181  
## 11 Sweden          1971  4431        7.38      327 0.252  
## 12 United Kingdom  1972  2742        4.45      122 0.0908 
## 13 United States   1972  5551        6.51      361 0.228

I guess Greece is kind of an outlier, but the biggest influencers here seem to be Sweden and the United States (with Norway picking up bronze, if you will). Let’s kick them out, see what happens.

summary(lm(medexppc ~ gdppc, data=subset(Newhouse77, country != "Norway")))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = subset(Newhouse77, country != 
##     "Norway"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -36.196 -17.929   3.192  18.096  32.135 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -63.679442  20.957463  -3.039   0.0125 *  
## gdppc         0.080917   0.006177  13.100 1.28e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.63 on 10 degrees of freedom
## Multiple R-squared:  0.9449, Adjusted R-squared:  0.9394 
## F-statistic: 171.6 on 1 and 10 DF,  p-value: 1.275e-07

summary(lm(medexppc ~ gdppc, data=subset(Newhouse77, country != "United States")))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = subset(Newhouse77, country != 
##     "United States"))
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -49.61 -15.85   5.53  20.39  31.39 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -72.287075  28.353316  -2.550   0.0289 *  
## gdppc         0.083029   0.008876   9.354 2.92e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.18 on 10 degrees of freedom
## Multiple R-squared:  0.8974, Adjusted R-squared:  0.8872 
## F-statistic: 87.51 on 1 and 10 DF,  p-value: 2.92e-06

summary(lm(medexppc ~ gdppc, data=subset(Newhouse77, country != "Sweden")))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = subset(Newhouse77, country != 
##     "Sweden"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -39.042 -13.212  -4.105  20.482  30.754 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -52.574043  22.279198   -2.36     0.04 *  
## gdppc         0.075242   0.006676   11.27 5.26e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 24.6 on 10 degrees of freedom
## Multiple R-squared:  0.927,  Adjusted R-squared:  0.9197 
## F-statistic:   127 on 1 and 10 DF,  p-value: 5.259e-07

Nothing materially seems to change. So, how about this to assess the model fit. We’ll do what’s called a “leave one out” approach here, or at least we’ll do it informally. We’ll write a simple script that just punts one observation out, repeat the regression, and we’ll present what happens to the effect of GDP per capita on per capita medical expenditures when that observation is removed.

countries <- Newhouse77$country

loopcountries <- function(dat){
  output <- tibble()
  for(i in countries) {
    dat %>%
      filter(country != i) -> dat1
    mod <- lm(medexppc ~ gdppc, data = dat1)
    broomed <- broom::tidy(mod) %>% mutate(omitted = i)
    output <- bind_rows(output, broomed)
  }
  return(output)
}

loopcountries(Newhouse77) %>%
  bind_rows(., broom::tidy(M1) %>% mutate(omitted = "FULL MODEL")) %>%
  mutate(lwr = estimate - abs(qnorm(.025))*std.error,
         upr = estimate + abs(qnorm(.025))*std.error) %>%
  # Let's get rid of the intercept here just to make this clearer
  filter(term != "(Intercept)") %>%
  ggplot(.,aes(as.factor(omitted), estimate, ymin=lwr, ymax=upr)) +
  facet_wrap(~term) +
  geom_pointrange() +
  geom_hline(yintercept = 0, linetype="dashed") +
  coord_flip()

Seems like the U.S. is more the influencer than Sweden or Greece, not that it materially changes much. Let’s get back on track with some diagnostics of this new M1 (from above). This is the full data, including Greece.

First, a residuals-fitted plot.

plot(M1, which=1)

This is mostly flat, but you’re asking a lot to get variation from 13 units. Oddly enough, the plot is complaining about Norway (makes sense), Sweden (makes sense), and the United Kingdom (which is odd, given the Cook’s D).

Here’ a Q-Q plot.

plot(M1, which=2)

The Q-Q plot has a jump in the middle there. Not too damning, but, again, asking a lot of 13 units.

Scale-location plot time:

plot(M1, which=3)

This scale-location plot has some weirdness happening. Again: 13 units.

Finally, the residuals-leverage plot.

plot(M1, which=5)

Oof, this really wants to call out the U.S. and Sweden as outliers, along with Norway. Well, fine. Let’s remove them and see what happens.

summary(lm(medexppc ~ gdppc, data=subset(Newhouse77, country != "United States" & country != "Sweden" & country != "Norway")))

## 
## Call:
## lm(formula = medexppc ~ gdppc, data = subset(Newhouse77, country != 
##     "United States" & country != "Sweden" & country != "Norway"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.168 -14.107   3.505  17.378  22.995 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -71.311648  25.916105  -2.752    0.025 *  
## gdppc         0.083326   0.008752   9.521 1.22e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 22.1 on 8 degrees of freedom
## Multiple R-squared:  0.9189, Adjusted R-squared:  0.9088 
## F-statistic: 90.65 on 1 and 8 DF,  p-value: 1.223e-05

I mean, sure, effect shifts up with them removed, but the estimate that still emerges is with a std.error of the full model.

Finally, let’s calculate elasticities from Table 3 and, goddamn, it’s been a hot minute since I’ve seen regression elasticities. I don’t think you’ll see this much in your travels. Indeed, the discussion here is very much in line with economics and the income elasticity of demand. Alas, let’s do it for max reproducibility.

tibble(gdppc = c(mean(Newhouse77$gdppc), 3416, 4000, 5000, 6000)) -> newdat # i.e. I smell an error since I know 3416 is not the actual mean.

# small little error. Basically: the first elasticity for each is correct but the mean is incorrectly labeled.
newdat %>%
  mutate(m1coef = M1$coefficients["gdppc"],
         m2coef = M2$coefficients["gdppc"],
         pred1 = predict(M1, newdat = .),
         pred2 = predict(M2, newdat = .),
         el1 = m1coef*(gdppc)/pred1,
         el2 = m2coef*(gdppc)/pred2)

## # A tibble: 5 x 7
##   gdppc m1coef m2coef pred1 pred2   el1   el2
##   <dbl>  <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 3261. 0.0788 0.0764  196.  198.  1.31  1.26
## 2 3416  0.0788 0.0764  209.  210.  1.29  1.24
## 3 4000  0.0788 0.0764  255.  254.  1.24  1.20
## 4 5000  0.0788 0.0764  333.  331.  1.18  1.15
## 5 6000  0.0788 0.0764  412.  407.  1.15  1.13

The results suggest that an implied income (regression) elasticity above 1 suggests medical care is a luxury good and not a normal good. I will, umm, not waste time on that particular point and its implications here.

Finally, here are the alternative estimations from Table 4. Of note: the I() wrapper in a regression is great for embedding on-the-fly transformations of the data. You don’t need it for on-the-fly log transformations. I recommend just creating additional columns.

summary(M3 <- lm(medsharegdp ~ I(1/gdppc), Newhouse77))

## 
## Call:
## lm(formula = medsharegdp ~ I(1/gdppc), data = Newhouse77)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.2086 -0.5323 -0.1928  0.6558  1.1422 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     8.1585     0.6415  12.717 6.39e-08 ***
## I(1/gdppc)  -6883.1614  1719.4624  -4.003  0.00208 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8464 on 11 degrees of freedom
## Multiple R-squared:  0.593,  Adjusted R-squared:  0.556 
## F-statistic: 16.02 on 1 and 11 DF,  p-value: 0.002075

summary(M4 <- lm(medsharegdp ~ gdppc, Newhouse77))

## 
## Call:
## lm(formula = medsharegdp ~ gdppc, data = Newhouse77)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.88684 -0.92375  0.06491  0.88324  1.14878 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)   
## (Intercept) 3.3978705  0.9005664   3.773  0.00308 **
## gdppc       0.0007271  0.0002623   2.772  0.01816 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.018 on 11 degrees of freedom
## Multiple R-squared:  0.4113, Adjusted R-squared:  0.3577 
## F-statistic: 7.684 on 1 and 11 DF,  p-value: 0.01816

summary(M5 <- lm(log(medsharegdp) ~ log(gdppc), Newhouse77))

## 
## Call:
## lm(formula = log(medsharegdp) ~ log(gdppc), data = Newhouse77)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.35799 -0.16611  0.03743  0.13048  0.28300 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -2.7032     1.2602  -2.145  0.05512 . 
## log(gdppc)    0.5510     0.1568   3.515  0.00484 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1991 on 11 degrees of freedom
## Multiple R-squared:  0.529,  Adjusted R-squared:  0.4862 
## F-statistic: 12.36 on 1 and 11 DF,  p-value: 0.004841

summary(M6 <- lm(medsharegdp ~ I(1/gdppc), data=subset(Newhouse77, country != "Greece")))

## 
## Call:
## lm(formula = medsharegdp ~ I(1/gdppc), data = subset(Newhouse77, 
##     country != "Greece"))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.4032 -0.4741  0.2248  0.6392  0.9988 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     7.2383     0.9097   7.957 1.23e-05 ***
## I(1/gdppc)  -3797.8563  2785.4186  -1.363    0.203    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8139 on 10 degrees of freedom
## Multiple R-squared:  0.1568, Adjusted R-squared:  0.07244 
## F-statistic: 1.859 on 1 and 10 DF,  p-value: 0.2026

OLS Regression

Steven V. Miller, svmiller.com

18 February 2021

R Packages/Data for This Session

OLS with Fake Data

The Basic Diagnostic Checks for Beginners

A Replication of Newhouse (1977)