Econometrics Problem Solver, Study Guides, Projects, Research of Econometrics and Mathematical Economics

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Thomas Andren

Econometrics

Econometrics

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Contents

Contents

1 Basics of probability and statistics 8 1.1 Random variables and probability distributions 8 1.2 The multivariate probability distribution function 15 1.3 Characteristics of probability distributions 17

2 Basic probability distributions in econometrics 24 2.1 The normal distribution 24 2.2 The t-distribution 31 2.3 The Chi-square distribution 33 2.4 The F-distribution 34

3 The simple regression model 36 3.1 The population regression model 36 3.2 Estimation of population parameters 41

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Contents

4 Statistical inference 49 4.1 Hypothesis testing 50 4.2 Confidence interval 52 4.3 Type I and type II errors 55 4.4 The best linear predictor 58

5 Model Measures 61 5.1 The coefficient of determination (R 2 ) 61 5.2 The adjusted coefficient of determination (Adjusted R2) 66 5.3 The analysis of variance table (ANOVA) 67

6 The multiple regression model 70 6.1 Partial marginal effects 70 6.2 Estimation of partial regression coefficients 72 6.3 The joint hypothesis test 73

7 Specification 78 7.2 Omission of a relevant variable 83 7.3 Inclusion of an irrelevant variable 85 7.4 Measurement errors 86

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Contents

11 Multicollinearity and diagnostics 129 11.1 Consequences 130 11.2 Measuring the degree of multicollinearity 133 11.3 Remedial measures 136

12 Simultaneous equation models 137 12.1 Introduction 137 12.4 Estimation methods 146

13 Statistical tables 152

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Econometrics

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Basics of probability and statistics

1 Basics of probability

and statistics

The purpose of this and the following chapter is to briefly go through the most basic concepts in probability theory and statistics that are important for you to understand. If these concepts are new to you, you should make sure that you have an intuitive feeling of their meaning before you move on to the following chapters in this book.

1.1 Random variables and probability distributions

The first important concept of statistics is that of a random experiment. It is referred to as any process of measurement that has more than one outcome and for which there is uncertainty about the result of the experiment. That is, the outcome of the experiment can not be predicted with certainty. Picking a card from a deck of cards, tossing a coin, or throwing a die, are all examples of basic random experiments.

The set of all possible outcomes of an experiment is called the sample space of the experiment. In case of tossing a coin, the sample space would consist of a head and a tail. If the experiment was to pick a card from a deck of cards, the sample space would be all the different cards in a particular deck. Each outcome of the sample space is called a sample point.

An event is a collection of outcomes that resulted from a repeated experiment under the same condition. Two events would be mutually exclusive if the occurrence of one event precludes the occurrence of the other event at the same time. Alternatively, two events that have no outcomes in common are mutually exclusive. For example, if you were to roll a pair of dice, the event of rolling a 6 and of rolling a double have the outcome (3,3) in common. These two events are therefore not mutually exclusive.

Events are said to be collectively exhaustive if they exhaust all possible outcomes of an experiment. For example, when rolling a die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes. Hence, the set of all possible die rolls is both mutually exclusive and collectively exhaustive. The outcomes 1 and 3 are mutually exclusive but not collectively exhaustive, and the outcomes even and not-6 are collectively exhaustive but not mutually exclusive.

Even though the outcomes of any random experiment can be described verbally, such as described above, it would be much easier if the results of all experiments could be described numerically. For that purpose we introduce the concept of a random variable. A random variable is a function that assigns unique numerical values to all possible outcomes of a random experiment.

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Basics of probability and statistics

Example 1.

You would like to know the probability of receiving 7 when rolling two dice. First we have to find the total number of unique outcomes using two dice. By forming all possible combinations of pairs we have (1,1), (1,2),…, (5,6),(6,6), which sum to 36 unique outcomes. How many of them sum to 7? We have (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): which sums to 6 combinations. Hence, the corresponding probability would therefore be 6/36 = 1/6.

The classical definition requires that the sample space is finite and that each outcome in the sample space is equally likely to appear. Those requirements are sometimes difficult to stand up to. We therefore need a more flexible definition that handles those cases. Such a definition is the so called relative frequency definition of probability or the empirical definition. Formally, if in n trials, m of them are favorable to the event A, then P(A) is the ratio m/n as n goes to infinity or in practice we say that it has to be sufficiently large.

Example 1.

Let us say that we would like to know the probability to receive 7 when rolling two dice, but we do not know if our two dice are fair. That is, we do not know if the outcome for each die is equally likely. We could then perform an experiment where we throw two dice repeatedly, and calculate the relative frequency. In Table 1.1 we report the results for the sum from 2 to 7 for different number of trials.

Number of trials Sum 10 100 1000 10000 100000 1000000 ∞ 2 0 0.02 0.021 0.0274 0.0283 0.0278 0. 3 0.1 0.02 0.046 0.0475 0.0565 0.0555 0. 4 0.1 0.07 0.09 0.0779 0.0831 0.0838 0. 5 0.2 0.12 0.114 0.1154 0.1105 0.1114 0. 6 0.1 0.17 0.15 0.1389 0.1359 0.1381 0. 7 0.2 0.17 0.15 0.1411 0.1658 0.1669 0.

Table 1.1 Relative frequencies for different number of trials

From Table 1.1 we receive a picture of how many trials we need to be able to say that that the number of trials is sufficiently large. For this particular experiment 1 million trials would be sufficient to receive a correct measure to the third decimal point. It seem like our two dices are fair since the corresponding probabilities converges to those represented by a fair die.

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Basics of probability and statistics

1.1.1 Properties of probabilities

When working with probabilities it is important to understand some of its most basic properties. Below we will shortly discuss the most basic properties.

1. 0 ≤ P ( A )≤ 1 A probability can never be larger than 1 or smaller than 0 by definition.
2. If the events A, B, … are mutually exclusive we have that P (^ A +^ B +...)= P ( A )+ P ( B )+...

Example 1.

Assume picking a card randomly from a deck of cards. The event A represents receiving a club, and event B represents receiving a spade. These two events are mutually exclusive. Therefore the probability of the event C = A + B that represents receiving a black card can be formed by P ( A + B )= P ( A )+ P ( B )

3. If the events A, B, … are mutually exclusive and collectively exhaustive set of events then we have that P ( A + B +...)= P ( A )+ P ( B )+...= 1

Example 1.

Assume picking a card from a deck of cards. The event A represents picking a black card and event B represents picking a red card. These two events are mutually exclusive and collectively exhaustive. Therefore P ( A + B )= P ( A )+ P ( B )= 1.

4. If event A and B are statistically independent then P(AB) = P(A)P(B) where P(AB) is called a joint probability.
5. If event A and B are not mutually exclusive then P (^^ A +^ B )= P ( A )+ P ( B )− PP(AB) ( AB )

Example 1.

Assume that we carry out a survey asking people if they have read two newspapers (A and B) a given day. Some have read paper A only, some have read paper B only and some have read both A and B. In order to calculate the probability that a randomly chosen individual has read newspaper A and/or B we must understand that the two events are not mutually exclusive since some individuals have read both papers. Therefore P ( A + B )= P ( A )+ P ( B )− P(AB)P ( AB ). Only if it had been an impossibility to have read both papers the two events would have been mutually exclusive.

Suppose that we would like to know the probability that event A occurs given that event B has already occurred. We must then ask if event B has any influence on event A or if event A and B are independent. If there is a dependency we might be interested in how this affects the probability of event A to occur. The conditional probability of event A given event B is computed using the formula:

PB

P A B = PP(AB)AB^ (1.2)

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Basics of probability and statistics

Example 1.

Consider a simple experiment where we toss a coin three times. Each trial of the experiment results in an outcome. The following 8 outcomes represent the sample space for this experiment: (HHH), (HHT), (HTH), (HTT), (THH), (THT), (TTH), (TTT). Observe that each sample point is equally likely to occure, so that the probability that one of them occure is 1/8.

The random variable we are interested in is the number of heads received on one trial. We denote this random variable X. X can therefore take the following values 0, 1, 2, 3, and the probabilities of occurrence differ among the alternatives. The table of probabilities for each value of the random variable is referred to as the probability distribution. Using the classical definition of probabilities we receive the following probability distribution.

X 0 1 2 3 P( X ) 1/8 3/8 3/8 1/

Table 1.3 Probability distribution for X

From Table 1.3 you can read that the probability that X = 0, which is denoted P ( X = 0 ), equals 1/8, whereas P ( X = 1 )equals 3/8, and so forth.

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Basics of probability and statistics

1.1.3 The cumulative probability function – the discrete case

Related to the probability mass function of a discrete random variable X, is its Cumulative Distribution Function , F(X), usually denoted CDF. It is defined in the following way:

F ( X )= P ( X ≤ c ) (1.3)

Example 1.

Consider the random variable and the probability distribution given in Example 1.8. Using that information we may form the cumulative distribution for X:

X 0 1 2 3 P( X ) 1/8 4/8 7/8 1

Table 1.4 Cumulative distribution for X

The important thing to remember is that the outcomes in Table 1.3 are mutually exclusive. Hence, when calculating the probabilities according to the cumulative probability function, we simply sum over the probability mass functions. As an example:

P ( X ≤ 2 )= P ( X = 0 )+ P ( X = 1 )+ P ( X = 2 )

1.1.4 The probability function – the continuous case

When the random variable is continuous it is no longer interesting to measure the probability of a specific value since its corresponding probability is zero. Hence, when working with continuous random variables, we are concerned with probabilities that the random variable takes values within a certain interval. Formally we may express the probability in the following way:

≤ ≤ = ∫

b

a

P ( a X b ) f ( x ) dxdx (1.4)

In order to find the probability, we need to integrate over the probability function, f(X), which is called the probability density function (pdf) for a continuous random variable. There exist a number of standard probability functions, but the single most common one is related to the standard normal random variable.

Example 1.

Assume that X is a continuous random variable with the following probability function:

=^ − >

else

f X e X X 0

( )^330

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Basics of probability and statistics

X 0 1 2 Total

Y

0 1/16 2/16 1/16 4/ 1 2/16 4/16 2/16 8/ 2 1/16 2/16 1/16 4/ Total 4/16 8/16 4/16 1.

Table 1.5 Joint probability mass function, (^) f ( X , Y )

As an example, we can read that P ( X = 0 , Y = 1 )= 2 2/16 = 1/8/ 16 = 1 / 8. Using this table we can for instance determine the following probabilities:

P ( X  Y ) P ( X 0 , Y 1 ) P ( X 0 , Y 2 ) P ( X 1 , Y 2 )^2  

P ( X! Y ) P ( X 1 , Y 0 ) P ( X 2 , Y 0 ) P ( X 2 , Y 1 )^2  

P ( X Y ) P ( X 0 , Y 0 ) P ( X 1 , Y 1 ) P ( X 2 , Y 2 )^1  

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Basics of probability and statistics

Using the joint probability mass function we may derive the corresponding univariate probability mass function. When that is done using a joint distribution function we call it the marginal probability function. It is possible to derive a marginal probability function for each variable in the joint probability function. The marginal probability functions for X and Y are

= ∑ y

f ( X ) f ( X , Y )for all X (1.6)

= ∑ x

f ( Y ) f ( X , Y )for all Y (1.7)

Example 1.

Find the marginal probability functions for the random variables X given in Table 1.5.

P ( X 0 ) f ( X 0 , Y 0 ) f ( X 0 , Y 1 ) f ( X 0 , Y 2 )^1  

P ( X 1 ) f ( X 1 , Y 0 ) f ( X 1 , Y 1 ) f ( X 1 , Y 2 )^2  

P ( X 2 ) f ( X 2 , Y 0 ) f ( X 2 , Y 1 ) f ( X 2 , Y 2 )^1  

Another concept that is very important in regression analysis is the concept of statistically independent random variables. Two random variables X and Y are said to be statistically independent if and only if their joint probability mass function equals the product of their marginal probability functions for all combinations of X and Y:

f ( X , Y )= f ( X ) f ( Y )for all X and Y (1.8)

1.3 Characteristics of probability distributions

Even though the probability function for a random variable is informative and gives you all information you need about a random variable, it is sometime too much and too detailed. It is therefore convenient to summarize the distribution of the random variable by some basic statistics. Below we will shortly describe the most basic summary statistics for random variables and their probability distribution.

1.3.1 Measures of central tendency

There are several statistics that measure the central tendency of a distribution, but the single most important one is the expected value. The expected value of a discrete random variable is denoted E[X], and is defined as follows:

E[[XY ]]= (^) ∑ X

n i

EX = (^) ∑ xi f xi = μ = 1

i = 1

n

Econometrics

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Basics of probability and statistics

The positive square root of the variance is the standard deviation and represents the mean deviation from the expected value in the population. The most important properties of the variance is

The variance of a constant is zero. It has no variability. If a and b are constants then (^) 9DU D;  E 9DUD; D  9DU^ ; Alternatively we have that Var(X) = E[X 2 ] – E[X]^2 E[X 2 ] = (^) ∑ x

x^2 f ( X )

Example 1.

Calculate the variance of X using the following probability distribution:

X 1 2 3 4 P( X ) 1/10 2/10 3/10 4/

Table 1.6 Probability distribution for X

In order to find the variance for X it is easiest to use the formula according to property 4 given above. We start by calculating E[X^2 ] and E[X].

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Basics of probability and statistics

 u u  u  u

(> ; @   

 u u  u  u 9DU> ; @ ± ^ 

1.3.3 Measures of linear relationship

A very important measure for a linear relationship between two random variables is the measure of the covariance. The covariance betwee X and Y is defined as

&RY > ;  < @ ( > ;  ( > ; @ <  (< @ ( > ;< @  ( > ; @ ( > @ < (1.12)

The covariance is the measure of how much two random variables vary together. When two variables tend to vary in the same direction, that is, when the two variables tend to be above or below their expected value at the same time, we say that the covariance is positive. If they tend to vary in opposite direction, that is, when one tends to be above the expected value when the other is below its expected value, we have a negative covariance. If the covariance is zero, we say that there is no linear relation between the two random variables.

Important properties of the covariance

1. (^) Cov [ X , X ] = Var [ X ]

2. Cov [ X , Y ] = Cov [ Y , X ]

4. (^) Cov [ X , Y + Z ] = Cov [ X , Y ] + Cov [ X , Z ]

The covariance measure is level dependent and has a range from minus infinity to plus infinity. That makes it very hard to compare two covariances between different pairs of variables. For that matter it is sometimes more convenient to standardize the covariance so that it become unit free and work within a much narrower range. One such standardization gives us the correlation between the two random variables.

The correlation between X and Y is defined as

[ ] Var [ X ] Var [ ] Y

Corr ( X , Y )= CovX , Y (1.13)

The correlation coefficient is a measure for the strength of the linear relationship and range from -1 to 1.