# Confidence Intervals in R

In this post, I will show how one can easily construct confidence intervals in R. Assume you have a vector of numbers and you want to construct a confidence interval around the mean of this vector. The subsequent R code shows one easy way to calculate the confidence interval around the mean of this vector. The following code loads a function that allows you to pass on the vector and returns the confidence intervals. Per default the function returns the 95% confidence interval. However, the parameter ‘conf_level’ allows you to specify the interval you want. Furthermore, if you do not have many observations, you may want to use Student’s t-distribution instead of the Normal distribution. The Student’s t-distribution has wider tales when the number of observations is low and gives a you more conservative estimates of your confidence interval. In case you want to use Student’s t-distribution you case set the parameter ‘distribution’, i.e. distribution=”normal”.

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# Generate Gamma Distributed Numbers in Julia

In Julia, one can generate random numbers that follow a Gamma distribution by using the Distribution package. Thereby one can use the rand() function that draws random numbers and specify the Gamma distribution by using the Gamma(a,b) command. The parameters a and b define the shape parameters of the Gamma distribution. This article provides a more generic overview of how to generate random numbers in Julia.

# Aggregate Demand

What is aggregate demand? Aggregate demand refers to total expenditure in an economy in a certain period. That is, aggregate demand comprises everything that is spend in an economy in one period. One can split aggregate demand into different subcomponents. Formally, one can describe aggregate demand (Y) as

Y = C + I + G + NX

As one can see from the equation above, aggregate demand (Y) is equal consumption (C) plus investment (I) plus government spending (G) plus net exports (NX), i.e. how much we are selling abroad to other countries on net.

According to Keynesian theory, aggregate demand determines the amount of available expenditure in an economy. Now, why should one care about available expenditure? Well, in Keynesian economics, available expenditure determines the amount of means available in an economy in order to sustain labor hires in a given period. That is, in the Keynesian model, the available expenditures is what keeps people at work. Boldly speaking, the amount of expenditure defines the amount of available money to pay the wages of workers. This concept is particularly important during a recession. Assume for instance, that a shock hits the economy and aggregate demand decreases. This implies that demand for firms’ products drops and firms will sell less products and earn less money. Hence, at the end of the month firms have less money available to pay their employees. Meaning that firms will be forced to lay off some workers and unemployment increases. Hence, in a Keynesian setting, a drop in aggregate demand implies a decrease in the means available in an economy, leading to less jobs and higher unemployment.

# Keynesian Economics

Keynesian Economics is a central doctrine in economics that forms the foundation of modern macroeconomic thinking. It was established by John Maynard Keynes during the 1930s. In 1936, the British economist published his book “The General Theory of Employment, Interest and Money” that forms the basis of the Keynesian school. The following series of blog posts will introduce Keynesian Economics, provide useful insights and discuss the most important concepts. The first post is dedicated to aggregate demand, the key concept of Keynesian Economics. The following post explains the dynamics of Keynesian Economics in an aggregate supply/ aggregate demand (AS-AD) model. The third post discusses the remedies of Keynesian Economics in the light of a recession. Finally, the last post elaborates some drawbacks of Keynesianism.

# Derivation of the Least Squares Estimator for Beta in Matrix Notation – Proof Nr. 1

In the post that derives the least squares estimator, we make use of the following statement:

$\frac{\partial b'X'Xb}{\partial b} =2X'Xb$

This post shows how one can prove this statement. Let’s start from the statement that we want to prove:

$\frac{\partial \hat{\beta}'X'X\hat{\beta}}{\partial \hat{\beta}}=2 X'X \hat{\beta}'$