Econometrics B/C

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# Econometrics B/C
## Lecture 1
### Jonas Björnerstedt
### 2021-09-29

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## B/C econometrics overview

- 10 Lectures

1. Statistics

1. Econometric theory

1. Use statistical software - R and Rstudio

- Empirical exercises

- Data management - create datasets
  
  - Data analysis - analysis of data
  
  - Regression analysis - formal analysis of data

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## Examination

- Written Exam 50 points

- Econometrics (exam) 2021-10-29 09 10.00 -14.00 __on campus__

- Econometrics (reexam) 2021-12-10 15.00 -19.00 __on campus__

- Empirical Exercises 50 points

- Three exercises, distributed during the course
  
  - Must be done in the virtual datalab
  
- 50 points to pass course

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## Textbook

- Online textbook

[Introduction to Econometrics with R](https://scpoecon.github.io/ScPoEconometrics/) - Florian Oswald, Jean-Marc Robin and Vincent Viers
  
  - Included in the course: Chapters 1-7 and 12.

- The course information included in the first section _Syllabus_ is not relevant for our course.
  
  - The content of chapters 1 and 2 are better covered in the two texts below
  
- R for data science - Hadley Wickham - [chapter 5](https://r4ds.had.co.nz/transform.html)

- A ModernDive into R and the Tidyverse - Chester Ismay and Albert Y. Kim - [Chapter 2](https://moderndive.com/2-viz.html) and [Chapter 3](https://moderndive.com/3-wrangling.html)

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## Web resources

#### Econometrics pages - sites.google.com/view/bc-econometrics

- Contains additional resources

- Lecture slides will be posted on the net

- Both lecture and exercise slides
  
#### Other resources

- Google for questions

- The internet!
    - Wikipedia is very good in probability and statistics

- [Khan academy](https://www.khanacademy.org/math/statistics-probability)
    - From basic to advanced with [an app with videos and exercises](https://itunes.apple.com/us/app/khan-academy-you-can-learn/id469863705?mt=8)

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# Today's lecture

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## Mean and variance

* Data in Y

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## Mean and variance

* Calculate the mean of Y

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## Mean and variance

* Calculate the deviation from the mean

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## Mean and variance

* Calculate the __square__ deviation from the mean (sq.dev)

* The __variance__ of Y is given by the _average_ square deviation

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# Apples

.pull-left[

- Pick apple from basket

- How much do you expect to eat?

]

.pull-right[

![](apples.jpeg)

]

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# Small and large apples

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Assume apples come from two different trees

- Small apples all weigh 100g

- Big apples all weigh 200g

- Same number of apples from each tree

]

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![](small-large-apple.jpg)

]

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## Expected value

- If you were given an apple, how much would you expect to eat?

- Either you eat 100g or 200g apple

- Big and small apples equally likely to be picked from the basket

- *In expecation* you eat 150g

- If `$Y$` is the amount you eat

- What is the expected value of `$Y$`?

- Let `$E(Y)$` denote the expected value of `$Y$`
`$$E(Y) = \frac{1}{2} 100 + \frac{1}{2} 200 = 150$$`
- Weighted average

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## Sampling Probability

- Probability

- Share of population with property
    
    - Share of random sample - frequency of event

- Population

- Example: Individuals in Sweden
    
    - Can be abstract set of states 
    
        - States of the world where a coin toss gives heads

- Sample

- Draws of individuals from population
    
    - Example: Class

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## Discrete random variable

- Finite discrete variable takes `$k$` different values

1. Length of individuals in a class

2. Outcomes of a coin toss

- Distribution can be characterized by the frequencies:

- Relative frequency of each age or length

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![](coin.gif)

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## Coin toss

- Coin toss has two outcomes (heads or tails)

- Assign a numerical value to each: -1, 1
    
    - Equal probability of each outcome with a _fair coin_

![](lecture01_files/figure-html/fig.width==1-1.png)

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## Discrete random variable - dice

- A toss of a die can have various outcomes

- Sample space: {1, 2, 3, 4, 5, 6}

- Each outcome occurs with equal probability

- Frequency with which we expect outcome

- _Probability Mass Function (PMF)_

- function that assigns a _probability_ to each outcome in the sample space
    
![](lecture01_files/figure-html/unnamed-chunk-6-1.png)

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## Probability and statistics

- Random variables

- Numerical properties of individuals
    
    - Examples: height, weight and gender

- Characterized by *probability distribution*

- Probability of each value that variable can take
    
    - Example: frequencies of all lengths in population

- Summarize with a *statistic*

- Real or vector valued _function_ of sample
    
    - Random variable (it depends on a random sample)
    
    - Sampling distribution of statistic?

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## Expected value

- A *Statistic* summarizes properties of distributions

- A real valued function of the probability distribution

- If `$Y$` has a discrete distribution:
    `$$E(Y)= \sum_i^k Y_i p_i =\mu_Y$$`

- For dice: 
`$$E(Y)=1*\frac{1}{6}+2*\frac{1}{6}+3*\frac{1}{6}+4*\frac{1}{6}+5*\frac{1}{6}+6*\frac{1}{6}=3.5$$`

- Populations often have equal weights

- Ex: The mean height of the Swedish population is just the average
    
    - Sum the weights of everybody and divide by the number of people

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## Variance

- The variance is a measure of the spread around the expected value

- How big is the square deviation on average?

- Let `$r$` be the square deviation:
`$$r = (Y - E[Y])^2$$`
- Then the variance is the expected value of the square deviation:
`$$Var(Y) = E[r] = E\left[(Y - E[Y])^2 \right]$$`
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## Conditional expectation

* Conditional expectation - expected value given something

* Two population variables `$length$` and `$woman$`

* Expected length of women: `$E(length | woman = 1 )$`

* Population average for women
  
  * `$woman$` is dummy variable, with values 0 and 1

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## Transform a random variable

- From a random variable `$Y$` we can create new random variables:

* `$2Y$` stretches
* `$Y + 1$` moves
* Squared deviation `$r$` is a random variable

* [Illustrate with coin toss]

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## Exercise

- Coin toss has two outcomes (heads or tails)

- Random variable taking values: -1, 1
    
    - What is the expected value and variance?

- Consider a random variable that assigns 0, 1 to outcomes

- What is the expected value and variance?

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## Mean and median of distribution

- _Mean_ - Average `$X$` value

- _Median_ - `$X$` with half of the density (area) to the left and right

- Differs when distribution is not symmetric (skewed)
    
    - Example income

![](lecture01_files/figure-html/unnamed-chunk-7-1.png)

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## Properties of expectations

`$$E(aY) = aE(Y)$$`

- Similarly, one can show that
`$$E(X + Y) = E(X) + E(Y)$$`
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## Properties of variance

`$$Var(aX) = a^2 Var(X)$$`

- For independent `$X$` and `$Y$`:
`$$Var(X + Y) = Var(X) + Var(Y)$$`