review of bas stats for econometric success

7/29/2019 review of bas stats for econometric success

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Econometrics

Review of Basic Statistics


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Topics

1. Descriptive Statistics:

- 1 variable: Mean and Variance

- 2 variables: Covariance, Correlation

2. Hypothesis Testing


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Descriptive Statistics


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Inferential Statistics

Involves:

- Estimation

- Hypothesis Testing

Purpose:

- Make decisions aboutpopulation characteristics


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Descriptive Statistics


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Mean

Measure of central tendency

Affected by extreme values

Formula:


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Median


Middle value in ordered series

- If odd n, mean of the 2 middle values

Value that splits the distribution into two halves

Not affected by extreme values

Raw Data: 17 16 21 18 13 16 12 11

Ordered: 11 12 13 16 16 17 18 21

Position: 1 2 3 4 5 6 7 8


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Mode


Value that occurs most often

Not Affected by Extreme Values There may be more than one mode

Raw Data: 17 16 21 18 13 16 12 11 Ordered: 11 12 13 16 16 17 18 21


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Sample Variance

Measure of Dispersion around the Mean

Formula:


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Sample Standard Deviation

Measure of Dispersion around the Mean

Has the same unit of measurement as the

variable itself

Formula:


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Radom Variables

Random variable: numerical summary of arandom outcome

1. Discrete: only a discrete set of possible values

=> summarized by probability distribution: list ofall possible values of the variable and the

probability that each value will occur.

2. Continuous: continuum of possible values

=> summarized by the probability density

function (pdf)


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Probability Distribution

1. List of pairs [ Xi, P(Xi) ]

Xi = Value of Random Variable (Outcome)

P(Xi) = Probability Associated with Value

2. 0 P(Xi) 1 - Mutually exclusive (no overlap)

3. P(Xi) = 1 - Collectively exhaustive (nothing

left out)


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Mean and Variance: Discrete Case

Mean, or Expected Value

Weighted Average of All Possible Values

E(X) = X= XiP(Xi) Variance

Weighted Average Squared Deviation about

the MeanE(X) = X= (Xi - X)

2P(Xi)


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Covariance

- measures joint variability ofXand Y

For discrete RVs,

Can take any value in the real numbers Depends on units of measurement (e.g., dollars, cents)

cov(X,Y) > 0 means thatXand Ytend to move together

when y is above its mean, x tends to be above its mean

cov(X,Y) < 0 means thatXand Ytend to move in opposite

directions

15

))((),cov(YX

YXEYX ),())((1

YXPYX

n

i

YiXi


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Correlation

A more convenient measure of the relationship betweenX

and Yis correlation since it is normalized to lie inside

[-1;1] interval.

-1 < corr(X,Y) < 1

If corr(X,Y) = 0, thenXand Yare uncorrelated.

If corr(X,Y) > 0 , thenXand Yare positively correlated.

If corr(X,Y) < 0, thenXand Yare negatively correlated.

16

YX

XY

YX

YX

YXcorr

)var()var(

),cov(

),(


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Note

Covariance and correlation measure only linear

dependence!

Example: Cov(X,Y)=0

Does not necessarily imply that y and x are

independent.

They may be non-linearly related.

But if X and Y are jointly normally distributed,

then they are independent.


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18

The correlationcoefficient

measures linearassociation


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The Mean and Variance of Sums of

Random Variables

19

)()()( YEXEYXE

),cov(2)var()var()var( YXYXYX

),cov(2)var()var()var( YXYXYX


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Continuous Probability Distributions:

Normal Distribution

The notation reads Xis Normally distributedwith mean and variance 2

The PDF for a normal RV is

The normal distribution has a familiar bell-shape.

The normal density is symmetric around its mean, and 95% of

probability density lies in the region .

20

),(~2

NX

2

22

)(2

1exp

2

1)(

xxf

)96.1,96.1(

X

Y


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Effects of Varying Parameters


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Infinite Number of Normal

Distribution Tables

Normal distributions differ by mean and

standard deviations

Each distribution would require its own table

Thats an infinite number of tables!


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Standard Normal Distribution IfXis a Normal RV with mean and variance 2

has a normal distribution with mean 0

and variance 1, or standard normal distribution.

X

Z


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Example


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Values of the std normal CDF, ,are tabulated in Appendix Table 1

To compute probabilities for a normal RV, it

must be standardized by subtracting its meanand dividing by standard deviation

Example: Suppose Y ~ N(2,16), and we need

P(Y


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Moments: Skewness, Kurtosis

skewness=

3

3

Y

Y

E Y

= measures asymmetry in a

distribution

The larger the skewness (by absolute value), the moreasymmetric is distributionskewness = 0: distribution is symmetric

skewness > ( 3: heavy tails (leptokurtotic), i.e. extreme events are

more likely to occur


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27


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Central Limit Theorem


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Important Continuous Distributions

All derived from normal

distribution

2 distribution: arises from

sum of squared normalrandom variables

t distribution: arises from

ratios of normal and 2

variables

F distribution: arises from

ratios of2variables.


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Hypothesis Testing


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Identifying Hypotheses

1. Formulate the question, e.g. test that thepopulation mean is equal to 3

2. State the question statistically (H0: = 3)

3. State its alternative statistically (H1: 3)

4. Choose level of significance

Typical values are 0.01, 0.05, 0.10

Rejection region of sampling distribution: the

unlikely values of sample statistic if null

hypothesis is true


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Identifying Hypotheses: Examples

1. Is the population average amount of TV

viewing 12 hours?

H0: = 12

H1: 12

Ch i th L l f Si ifi


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Choosing the Level of Significance:

Type I and Type II Errors

Type I Error: Reject a true null hypothesis.

Type II Error: Do not reject a false null.

We would like probabilities of both errors to be

small. BUT, we cannot make both very small at

the same time.

In statistics, we fix the probability of Type I errorat a significance level (e.g. 5%) and minimize the

probability of Type II error.


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Hypothesis Testing: Basic Idea


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Method 1: Compare Test Statistic to

Critical Value from the Table1. Convert Sample Statistic (e.g., ) to standardized

Z variable

2. Compare to Critical Value from the table

If Z-test statistic falls in the rejection region,reject H0;

Otherwise do not reject H0


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Two-Sided Test: Rejection Regions


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One-Sided Test: Rejection Region


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Method 2: Compute the P-value

Probability of obtaining a test statistic more

extreme ( or ) than actual sample value given

H0 is true.

The lowest significance level at which we reject H0

Compute p-value for the test

Use this p-value to make rejection decision: Ifp value , do not reject H0

Ifp value < , reject H0

l f 2 id d d 1 id d


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P-values for 2-sided and 1-sided tests

Two-sided test: H0: = 0

H1: 0

One-sided test:

a. H0: > 0 b. H0: < 0

H1: < 0 H1: > 0

YY

YYZPvaluep

00 2||

Y

Yvaluep

01

Y

Yvaluep

0


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P-value for a 2-Sided Test


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Method 3: Confidence Intervals

Confidence interval: set of values that containsthe true population mean with a pre-specified

probability, say 95%.

This pre-specified probability is called confidence

level.

A 95% confidence interval for Y contains the true

value ofY

in 95% of repeated samples.

The 90% CI is:

The 95% CI is:

The 99% CI is:41

)(96.1 YSEY

)(58.2 YSEY

)(645.1 YSEY


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Jarque-Bera Test for Normality

Assesses whether a given sample of data isnormally distributed

Aggregates information in the data about both

skewness and kurtosis Test of the hypothesis that S = 0 and K = 3

The 5% critical value is 5.99; if JB > 5.99, reject

the null of normality.

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