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ECON 351 - Fundamentals of
Mathematical Statistics
Maggie Jones
1 / 43
Populations and Sampling
I In econometrics our objective is to learn something about apopulation given a specific sample of that population
I Let Y be a random variable representing a population withpdf f(y; θ)
I If Y1, Y2, . . . , Yn are independent random variables with acommon pdf f(y; θ) then {Y1, Y2, . . . , Yn} is said to be arandom sample from f(y; θ)
I also called identically and independently distributed
2 / 43
Finite Sample Properties of
Estimators
I The idea behind finite sample properties is that they hold for asample of any size - also known as small sample properties
I Given a random sample {Y1, Y2, . . . , Yn} drawn from apopulation distribution that depends on an unknownparameter θ, an estimator of θ is a rule that assigns eachpossible outcome of the sample a value of θ
I e.g., the sample average is computed as
Y =1
n
n∑i=1
Yi
3 / 43
Example of Estimator
I Each time we draw a new sample, the estimate of µ will beslightly different
I The distribution of Y is known as the sampling distribution
I There are many different ways to compute parameterestimates, we will eventually be studying the samplingdistributions of estimators
4 / 43
Unbiased Estimator
I An estimator W of θ is an unbiased estimator if E(W ) = θfor all values of θ
I If W is biased, we can compute the bias as
Bias(W ) ≡ E(W )− θ
I Define an estimator of Var(Y ) = σ2 as
S2 =1
n− 1
n∑i=1
(Yi − Y )2
I Is it unbiased?
5 / 43
Efficient Estimator
I The variance of an estimator is called the sampling variance
I e.g., the variance of the sample average for estimating themean µ from a population is
Var(Y ) = Var(1
n
n∑i=1
Yi) =σ2
n
I In choosing an efficient estimator we focus on the estimatorthat yields the smallest sampling variance
I Formally, if W1 and W2 are two unbiased estimators of θ, W1
is efficient relative to W2 if Var(W1) ≤ Var(W2) for all θ andwith strict inequality for at least one θ.
6 / 43
Efficient Estimator
I The efficiency criterion works well to assess estimators whenthey are unbiased
I In the case of unbiased estimators, we can use the meansquared error to compare estimators
I The mean squared error of an estimator of θ, W , is
MSE(W ) = E[(W − θ)2
]I The idea is to measure how far away the estimator is from θ
on average
7 / 43
Asymptotic Properties of Estimators
I Asymptotic, sometimes called large sample, properties describethe behaviour of estimators as the sample size getsincreasingly large
I In some cases, for instance, an estimator can be biased, butcan still have reasonable large sample properties
I “Large” is an ambiguous term - no formal rule for sample sizerequired to invoke asymptotic theory
8 / 43
Consistency
I Consistency tells whether the estimator converges to theparameter it is supposed to be estimating as the sample sizeincreases indefinitely
I Formally, let Wn be an estimator of θ based on a sampleY1, Y2, . . . , Yn of size n. Then, Wn is a consistent estimatorof θ if for every ε > 0
Pr(|Wn − θ|> ε)→ 0 as n→∞
I If Wn is consistent, it is also said that θ is the probabilitylimit of Wn
plim(Wn) = θ
9 / 43
Consistency for n = 10
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0 2000 4000 6000 8000 10000
9.0
9.5
10.0
10.5
11.0
Convergence in Probability
xbar
10 / 43
Consistency for n = 100
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0 2000 4000 6000 8000 10000
9.0
9.5
10.0
10.5
11.0
Convergence in Probability
xbar
11 / 43
Consistency for n = 1000
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0 2000 4000 6000 8000 10000
9.0
9.5
10.0
10.5
11.0
Convergence in Probability
xbar
12 / 43
Consistency
I Note that an unbiased estimator may not be consistent
I If the variance of an unbiased estimator shrinks as the samplesize grows, then it is consistent
I Formally, if Wn is an unbiased estimator of θ andVar(Wn)→ 0 as n→∞, then plim(Wn) = θ
I e.g., Is the average of a random sample drawn from apopulation with mean µ and variance σ2 consistent?
13 / 43
Law of Large Numbers
I The law of large numbers states that if Y1, Y2, . . . , Yn areindependent, identically distributed random variables withmean µ, then
plim(Yn) = µ
I If we are interested in estimating the population average µ, wecan get arbitrarily close to µ by choosing a sufficiently largesample.
14 / 43
Asymptotic Normality
I Consistency tells us that the distribution of the estimator iscollapsing around the parameter as the sample size gets large
I Sometimes we are more interested in the shape of thedistribution of the estimator for a given sample size
I Most of the estimators that we will consider are wellapproximated by a normal distribution
I A sequence of random variables {Zn : n = 1, 2, . . . } is said tohave an asymptotic standard normal distribution if forall numbers z
Pr(Zn ≤ z)→ Φ(z) as n→∞
15 / 43
Central Limit Theorem
I The central limit theorem states that if {Y1, Y2, . . . , Yn} is arandom sample with mean µ and variance σ2 then,
Zn =Yn − µσ/√n
has an asymptotic standard normal distribution
16 / 43
General Approaches to Parameter
Estimation
I Method of Moments:
I Maximum Likelihood:
I Least Squares:*
17 / 43
Interval Estimation and Confidence
Intervals
I How do we know if a parameter estimate is “reliable” in thesense that it provides information about the true parameter wewish to estimate?
I If we don’t know the true value, it is impossible to answer thisquestion for certain, but we can make statements involvingprobabilities
I We use confidence intervals to make statements aboutwhere the population value is likely to lie in relation to theestimate
18 / 43
Confidence Intervals
I Suppose a population is distributed normally with mean µ andvariance 1
I Let {Y1, . . . , Yn} be a random sample from this population
I The sample average, Y has a normal distribution with mean µand variance 1/n
19 / 43
Confidence Intervals
I Then,
Pr
(−1.96 <
Y − µ1/√n< 1.96
)= 0.95
which is equivalent to
Pr(Y − 1.96/
√n < µ < Y + 1.96/
√n)
= 0.95
I This is an example of an interval estimate of µ
I It tells us the probability that the random interval[Y − 1.96/
√n, Y + 1.96/
√n] contains µ is 95%
20 / 43
A Note on Interpreting Confidence
Intervals
I A confidence interval does not tell us that the probabilitythat µ is in the interval is 95%
I It tells us that for 95% of all random samples, the constructedconfidence interval will contain µ
21 / 43
Confidence Intervals
I In general, for a population with a tn−1 distribution that hasmean µ and variance σ, the 100(1− α)% confidence interval is:[
y − cα/2s√n, y + 1.96cα2s
√n]
22 / 43
Hypothesis Testing
I Suppose we have two candidates: A and B
I A receives 42% of the popular vote and B receives 58%
I A suggests the election was rigged and employs a pollingcompany to sample the population
I The polling company samples 100 people and finds that 53 ofthem voted for A
I Is the election rigged?
23 / 43
Hypothesis Testing
I Is the true proportion of people voting for candidate A 42%?
H0 : θ = 42︸ ︷︷ ︸Null Hypothesis
H1 : θ > 42︸ ︷︷ ︸Alternative Hypothesis
24 / 43
Hypothesis Testing
I Type 1 Error: We reject the null hypothesis when it is trueI e.g. reject H0 when the true proportion of people voting for A
is 42%
I Type 2 Error: We fail to reject the null hypothesis when it isfalse
I e.g. fail to reject H0 when the true proportion of people votingfor A is > 42%
25 / 43
Hypothesis Testing - Type 1 Errors
I We need a way to determine the likelihood of committing anerror (recall, we can never know for certain whether an errorhas been committed)
I We define the significance level (α) as the probability of aType 1 error:
α = Pr(Reject H0|H0)
I In words: the probability that we reject H0 given that H0 istrue
I We choose the significance level ahead of time, which can beinterpreted as our tolerance for a Type 1 error
26 / 43
Hypothesis Testing - Type 2 Errors
I Given the significance level, we construct our test such that weminimize the probability of committing a Type 2 error
I This is known as maximizing the power of a test against allrelevant alternatives
π(θ) = Pr(Reject H0|θ) = 1− Pr(Type 2|θ)
I In words: for a given alternative hypothesis, the probabilitythat the test correctly rejects the null hypothesis
27 / 43
Hypothesis Testing
I To perform the statistical test, we need to choose a teststatistic, T , which is a function of the random sample
I Denote the realization of T given our sample, t
I Given t, we must determine the rejection rule that specifieswhen the null hypothesis is rejected in favour of thealternative hypothesis
I The rejection rule is based on comparing the value of t to acritical value
I The values of t that result in rejection are collectively knownas the rejection region
28 / 43
Hypothesis Testing
I Suppose we wish to test a hypothesis about the mean µ fromN(µ, σ2)
I H0 : µ = µ0
I H1 : µ > µ0
I H1 : µ < µ0
I H1 : µ 6= µ0
I The alternative hypotheses are either one-sided or two-sided
I Our example above (is the election rigged?) is a one-sidedalternative
I The rejection rule we choose depends on the nature of thealternative hypothesis
29 / 43
Hypothesis Testing
I Note: Hypotheses are always written with respect to the truepopulation (e.g. µ), but are tested with respect to the samplewe have at hand (e.g. y)
30 / 43
Hypothesis Testing
I To determine whether we should reject H˙0 in favour of H˙1 weneed to determine the probability of rejecting the nullhypothesis when it is true.
I Typically we do not work directly with y, but we use itsstandardized version:
t =y − µ0
s/√n
I Since the random variable T = Y−µ0S/√n∼ tn−1 we will be
comparing our test statistic to the t distribution with n− 1degrees of freedom
31 / 43
Hypothesis Testing
I The test statistic t = y−µ0s/√n
is called the t-statistic for testingH0 : µ = µ0
I It measures the distance from y to µ0 relative to the standarderror of y, se(y)
32 / 43
Hypothesis Testing
I Suppose we settle on a 5% significance level.
I We choose the critical value, c so that the probability of aType 1 error is 5%
Pr(T > c|H0) = 0.05
I Where c is the 100(1-α) percentile in a tn−1 distribution
33 / 43
Hypothesis Testing - Rejection Rules
I Suppose we wish to test a hypothesis about the mean µ fromN(µ, σ2)
I H0 : µ = µ0
I H1 : µ > µ0
I H1 : µ < µ0
I H1 : µ = µ0
I The alternative hypotheses are either one-sided or two-sided
I Our example above (is the election rigged?) is a one-sidedalternative
I The rejection rule we choose depends on the nature of thealternative hypothesis
34 / 43
Hypothesis Testing - One-Sided
Rejection Region
0.1
.2.3
.4
-4 -2 0 2 4
35 / 43
Hypothesis Testing - Two-Sided
Rejection Region
0.1
.2.3
.4
-4 -2 0 2 4
36 / 43
Hypothesis Testing - Rejection Rules
I For the null hypothesis, H0 : µ = µ0, we have the followingrejection rules
I Reject H0 in favour of H1 if t > c if H1 : µ > µ0
I Reject H0 in favour of H1 if t < −c if H1 : µ < µ0
I Reject H0 in favour of H1 if |t|> c if H1 : µ = µ0
37 / 43
Hypothesis Testing
I Note: When a statistical test leads us to not reject the nullhypothesis we typically say that we, “fail to reject the nullhypothesis”. It is incorrect to say, “we accept the nullhypothesis,” because technically there are multiple nullhypotheses that may result in a failure to reject the null, yetthere is only one true parameter value.
38 / 43
Hypothesis Testing
I Note: If the sample size is large enough to invoke the CLTthen
T =Y − µ0
S/√n∼ N(0, 1)
I In these cases we can compare the t-statistic to the criticalvalue found from the standard normal distribution
39 / 43
Hypothesis Testing - the p-value
I A similar, but often more informative way to examinehypothesis testing is through the use of a p-value
I The p-value tells us the largest significance level at which wecould carry out the test and still fail to reject the null
40 / 43
Hypothesis Testing - the p-value
I The p-value is computed by inverting the test statistic that wecompute from the data:
p = Pr(T > 1.52|H0) = 1− Φ(1.52) = 0.065
I For a two sided test we need to account for the two sidednature of the test:
p = Pr(|t|> |c||H0) = 1− 2Φ(|t|)
I Where, as before, Φ is the cdf of the test statistic
41 / 43
Hypothesis Testing
I We can summarize the rejection rules based on test statisticsas:
I Reject H0 in favour of H1 if |t|> |c|
I Similarly, we can use the information in a p-value as arejection rule compared to the significance level, α:
I Reject H0 in favour of H1 if p < α
42 / 43
A Note on Interpretation
I How do we know if an estimate is meaningful?
I Distinguish between economic or practical significanceversus statistical significance
I e.g. Suppose you estimate that a post-secondary fundingprogram increases educational attainment by 0.000002% andthat the test for whether this estimate is different from zeroyields a p-value of 0.01
I Is this estimate statistically significant at the 5% level?I Is this estimate economically significant?I Would you implement this policy?
I Sometimes you need to use your judgement in econometrics
43 / 43
