some fixes

Author: mca91

.texpadtmp/ITER.aux

@@ -0,0 +1,45 @@
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.texpadtmp/ITER.log

@@ -0,0 +1,9 @@
+\contentsline {chapter}{Preface}{5}{chapter*.2}%
+\contentsline {subsubsection}{Conventions Used in this Book}{5}{section*.3}%
+\contentsline {subsubsection}{Acknowledgement}{6}{section*.4}%
+\contentsline {chapter}{\numberline {1}Introduction}{7}{chapter.1}%
+\contentsline {section}{\numberline {1.1}Colophon}{9}{section.1.1}%
+\contentsline {section}{\numberline {1.2}A Very Short Introduction to \texttt {R} and \emph {RStudio}}{13}{section.1.2}%
+\contentsline {subsubsection}{\texttt {R} Basics}{13}{section*.5}%
+\contentsline {subsubsection}{Vectors}{14}{section*.6}%
+\contentsline {subsubsection}{Functions}{15}{section*.7}%

.texpadtmp/ITER.synctex.gz

@@ -46,7 +46,6 @@ knitr::kable(pdfdata, format = my_output, caption = "PDF and CDF of a Dice Roll"
 
 We can easily plot both functions using `r ttcode("R")`. Since the probability equals $1/6$ for each outcome, we set up the vector `r ttcode("probability")` by using the function `r ttcode("rep()")` which replicates a given value a specified number of times.
 
-<div class="unfolded"> 
 ```{r, eval = T, message = F, warning = F, fig.align='center', fig.pos="h"} 
 # generate the vector of probabilities 
 probability <- rep(1/6, 6) 
@@ -56,11 +55,9 @@ plot(probability,
      xlab = "outcomes",
      main = "Probability Distribution") 
 ``` 
-</div>
 
 For the cumulative probability distribution we need the cumulative probabilities, i.e., we need the cumulative sums of the vector `r ttcode("probability")`. These sums can be computed using `r ttcode("cumsum()")`.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # generate the vector of cumulative probabilities 
 cum_probability <- cumsum(probability) 
@@ -70,7 +67,6 @@ plot(cum_probability,
      xlab = "outcomes", 
      main = "Cumulative Probability Distribution") 
 ```
-</div>
 
 ### Bernoulli Trials {-}
 
@@ -449,15 +445,15 @@ f(x) = \frac{1}{\sqrt{2 \pi} \sigma} \exp{-(x - \mu)^2/(2 \sigma^2)}.
 For the standard normal distribution we have $\mu=0$ and $\sigma=1$. Standard normal variates are often denoted by $Z$. Usually, the standard normal PDF is denoted by $\phi$ and the standard normal CDF is denoted by $\Phi$. Hence,
 $$ \phi(c) = \Phi'(c) \ \ , \ \ \Phi(c) = P(Z \leq c) \ \ , \ \ Z \sim \mathcal{N}(0,1).$$ Note that the notation X $\sim$ Y reads as "X is distributed as Y". In `r ttcode("R")`, we can conveniently obtain densities of normal distributions using the function `r ttcode("dnorm()")`. Let us draw a plot of the standard normal density function using `r ttcode("curve()")` together with `r ttcode("dnorm()")`.
 
-<div class="unfolded"> 
+
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # draw a plot of the N(0,1) PDF
 curve(dnorm(x),
       xlim = c(-3.5, 3.5),
       ylab = "Density", 
       main = "Standard Normal Density Function") 
 ```
-</div>
+
 
 We can obtain the density at different positions by passing a vector to `r ttcode("dnorm()")`.
 
@@ -468,15 +464,14 @@ dnorm(x = c(-1.96, 0, 1.96))
 
 Similar to the PDF, we can plot the standard normal CDF using `r ttcode("curve()")`. We could use `r ttcode("dnorm()")` for this but it is much more convenient to rely on `r ttcode("pnorm()")`.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'}
 # plot the standard normal CDF
 curve(pnorm(x), 
       xlim = c(-3.5, 3.5), 
       ylab = "Probability", 
       main = "Standard Normal Cumulative Distribution Function")
 ```
-</div>
+
 
 We can also use `r ttcode("R")` to calculate the probability of events associated with a standard normal variate. 
 
@@ -644,7 +639,6 @@ it holds that
 $$ Z_1^2+Z_2^2+Z_3^3 \sim \chi^2_3. \tag{2.3} $$
 Using the code below, we can display the PDF and the CDF of a $\chi^2_3$ random variable in a single plot. This is achieved by setting the argument `r ttcode("add = TRUE")` in the second call of `r ttcode("curve()")`. Further we adjust limits of both axes using `r ttcode("xlim")` and `r ttcode("ylim")` and choose different colors to make both functions better distinguishable. The plot is completed by adding a legend with help of `r ttcode("legend()")`.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # plot the PDF
 curve(dchisq(x, df = 3), 
@@ -666,15 +660,14 @@ legend("topleft",
        col = c("blue", "red"), 
        lty = c(1, 1))
 ```
-</div>
+
 
 Since the outcomes of a $\chi^2_M$ distributed random variable are always positive, the support of the related PDF and CDF is $\mathbb{R}_{\geq0}$.
 
 As expectation and variance depend (solely!) on the degrees of freedom, the distribution's shape changes drastically if we vary the number of squared standard normals that are summed up. This relation is often depicted by overlaying densities for different $M$, see the <a href="https://en.wikipedia.org/wiki/Chi-squared_distribution">Wikipedia Article</a>.  
 
 We reproduce this here by plotting the density of the $\chi_1^2$ distribution on the interval $[0,15]$ with `r ttcode("curve()")`. In the next step, we loop over degrees of freedom $M=2,...,7$ and add a density curve for each $M$ to the plot. We also adjust the line color for each iteration of the loop by setting `r ttcode("col = M")`. At last, we add a legend that displays degrees of freedom and the associated colors.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # plot the density for M=1
 curve(dchisq(x, df = 1), 
@@ -698,7 +691,7 @@ legend("topright",
        lty = 1, 
        title = "D.F.")
 ```
-</div>
+
 
 Increasing the degrees of freedom shifts the distribution to the right (the mode becomes larger) and increases the dispersion (the distribution's variance grows).
 
@@ -722,7 +715,6 @@ A $t_M$ distributed random variable $X$ has an expectation if $M>1$ and it has a
 
 Let us plot some $t$ distributions with different $M$ and compare them to the standard normal distribution.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # plot the standard normal density
 curve(dnorm(x), 
@@ -756,7 +748,6 @@ legend("topright",
        col = 1:4, 
        lty = c(2, 1, 1, 1))
 ```
-</div>
 
 The plot illustrates what has been said in the previous paragraph: as the degrees of freedom increase, the shape of the $t$ distribution comes closer to that of a standard normal bell curve. Already for $M=25$ we find little difference to the standard normal density. If $M$ is small, we find the distribution to have heavier tails than a standard normal, i.e., it has a "fatter" bell shape.
 
@@ -777,7 +768,6 @@ pf(2, df1 = 3, df2 = 14, lower.tail = F)
 
 We can visualize this probability by drawing a line plot of the related density and adding a color shading with `r ttcode("polygon()")`.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'}
 # define coordinate vectors for vertices of the polygon
 x <- c(2, seq(2, 10, 0.01), 10)
@@ -793,7 +783,7 @@ curve(df(x ,3 ,14),
 # draw the polygon
 polygon(x, y, col = "orange")
 ```
-</div>
+
 
 The $F$ distribution is related to many other distributions. An important special case encountered in econometrics arises if the denominator degrees of freedom are large such that the $F_{M,n}$ distribution can be approximated by the $F_{M,\infty}$ distribution which turns out to be simply the distribution of a $\chi^2_M$ random variable divided by its degrees of freedom $M$,
 
@@ -894,7 +884,6 @@ VarS
 
 So the distribution of $S$ is known. It is also evident that its distribution differs considerably from the marginal distribution, i.e,the distribution of a single dice roll's outcome, $D$ . Let us visualize this using bar plots.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # divide the plotting area into one row with two columns
 par(mfrow = c(1, 2))
@@ -919,8 +908,6 @@ barplot(probability,
         space = 0, 
         main = "Outcome of a Single Dice Roll")
 ```
-</div> 
-
 
 Many econometric procedures deal with averages of sampled data. It is typically assumed that observations are drawn randomly from a larger, unknown population. As demonstrated for the sample function $S$, computing an average of a random sample has the effect that the average is a random variable itself. This random variable in turn has a probability distribution, called the sampling distribution. Knowledge about the sampling distribution of the average is therefore crucial for understanding the performance of econometric procedures.
 
@@ -994,7 +981,6 @@ A straightforward approach to examine the distribution of univariate numerical d
 
 Using `r ttcode("curve()")`, we overlay the histogram with a red line, the theoretical density of a $\mathcal{N}(0, 0.1)$ random variable. Remember to use the argument `r ttcode("add = TRUE")` to add the curve to the current plot. Otherwise `r ttcode("R")` will open a new graphic device and discard the previous plot!^[*Hint:* `r ttcode("T")` and `r ttcode("F")` are alternatives for `r ttcode("TRUE")` and `r ttcode("FALSE")`.]  
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'}
 # Plot the density histogram
 hist(sample.avgs, 
@@ -1009,7 +995,6 @@ curve(dnorm(x, sd = 1/sqrt(n)),
       lwd = "2", 
       add = T)
 ```
-</div>
 
 The sampling distribution of $\overline{Y}$ is indeed very close to that of a $\mathcal{N}(0, 0.1)$ distribution so the Monte Carlo simulation supports the theoretical claim. 
 
@@ -1023,7 +1008,6 @@ To visualize the claim stated in equation (<a href="#mjx-eqn-2.3">2.3</a>), we p
 
 Again, we produce a density estimate for the distribution underlying our simulated data using a density histogram and overlay it with a line graph of the theoretical density function of the $\chi^2_3$ distribution.
 
-<div class="unfolded"> 
 ```{r, echo = T, eval = T, message = F, warning = F, fig.align='center'} 
 # number of repetitions
 reps <- 10000
@@ -1052,7 +1036,7 @@ curve(dchisq(x, df = DF),
       col = "red", 
       add = T)
 ```
-</div>
+
 
 ### Large Sample Approximations to Sampling Distributions {-}