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METRICS TERM 1 > HANDOUT 12 > Flashcards

HANDOUT 12 Flashcards

1
Q

4 different names for models when Y is not continuous

A
  1. Limited dependent variable models
  2. Binary choice models
  3. Dummy dependent variable
  4. Qualitative choice
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2
Q

What is the observed variable?

A 
Yi = 1 if vote
Yi = 0 if do not vote
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3
Q

What is the latent variable?

A

Yi* = net-utility from undertaking the activity

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4
Q

Y* =

A

Y* = Xi’B + €i in short form

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5
Q

What is the problem with Y* the latent variable?

A

It is UNOBSERVED

- we do not know an individual’s net utility from undertaking an action

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6
Q

How do we related Yi and Yi*?

A 
Yi = 1 if Yi* >=0
Yi = 0 if Yi* < 0
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7
Q

E(yi) based on bernoulli trial

A

E(yi) = p(Yi = 1) = pi

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8
Q

V(Yi) based on bernoulli trial

A

V(Yi) = pi (1-pi) = P(Yi = 1) x P(Yi = 0)

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9
Q

How can we rewrite E(Yi)?

A

E(Yi) = P(Yi=1) = P(Yi* >=0) = P(Xi’B + €i >=0)

= P(€i >= -Xi’B) = P(€i <= Xi’B) = F(Xi’B)

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10
Q

Distribution of €i

A

Normal distribution - symmetric

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11
Q

F(Xi’B) refers to

A

The cumulative distribution function - probability of being less than or equal to Xi’B under the distribution of €i

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12
Q

Our Model in 2 equations

A
  1. E(yi) = F(Xi’B)
  2. Yi = E(Yi) + Ui
    - this is always the case: y = its expected value + some error term
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13
Q

What is F in a linear probability model?

A 
F = a UNIFORM distribution
F(Xi'B) = U(L, U)
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14
Q

3 facts about uniform distribtuion

A
  1. centered at zero
  2. distributed between lower and upper limit
  3. all shocks equally likely
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15
Q

Under a uniform distribution, what is F(Xi’B0?

A

F(Xi’B) = Xi’B

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16
Q

Therefore, what is our model for LPM & how do we estimate it?

A

E(Yi) = F(Xi’B) = Xi’B
So: Yi = Xi’B + Ui
Estimate by usual OLS
Unless Xi endogenous –> IV estimation

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17
Q

Interpret coefficient on X1 under LPM

A

B1 = change in P(Y=1) for a unit increase in X1, ceteris paribus.

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18
Q

100B1 under LPM=

A

100B1 = percentage point change in P(Y=1) for 1 unit increase in X1

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19
Q

B1 if X1 is a dummy variable under LPM

A

B1 = change in P(Y=1) for having the characteristic vs not having it, ceteris paribus

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20
Q

3 advantages of LPM

A
  1. easy to estimate - OLS
  2. easy to interpret coefficients
  3. easy to solve endogeneity issue - IV
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21
Q

3 problems with LPM

A
  1. Ui is not normal
  2. Ui is heteroscedastic
  3. Pi is NOT bounded [0, 1]
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22
Q

Why is Ui not normal under LPM?

A

If Yi=0, Ui = -Xi’B
If Yi = 1, Ui = 1 - Xi’B
Only takes 2 values = cannot be normal

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23
Q

Is non-normality of Ui under LPM an issue?

A

NO - invoke CLT if n>=30

coefficients approx normal = do z tests and chi-squared tests

24
Q

V(Ui) =

A

V(Ui) = Xi’B(1 - Xi’B)

- Depends on i = heteroscedastic

25
Q

Is heteroscedasticity of Ui under LPM an issue?

A

NO - just use robust standard errors.

26
Q

Is Pi not bounded under LPM an issue?

A

YES - not well-defined

Pi = Xi’B - we cannot bound this between 0 and 1.

27
Q

Logit model what is F + formula

A 
F = logistic distribution
F(.) = e^. / (1 + e^.)
28
Q

Are probabilities bounded for logistic distribution?

A

YES - as Xi’B to infinity, F –>1
As Xi’B –> -infinity, F –> 0
As Xi’B –> 0, F–>1/2

29
Q

How do we estimate a Logit model?

A

MAXIMUM LIKELIHOOD ESTIMATION

30
Q

What does maximum likelihood estimation do? Coin flipping example.

A 
Suppose we flip a coin 30 times and observe 18 heads. We then try to find the p(head) that maximises the chance of what we observed.
Repeated bernoulli trials = binomial
P(X=18) = 30C18 x p^18 x (1 - p)^12
max w.r.t p
we get p* = 18/30 = 0.6
31
Q

If the sample is random, how can we write joint probabilities?

A

Just multiple the individual probabilities together

P(A n B) = P(A) x P(B)

32
Q

Denote the joint density function as the likelihood function

A

L(.) = Pi i=1,…,n [F(Xi’B)]^Yi [1 - F(Xi’B)]^1-Yi

33
Q

Take logs of likelihood function

A

ln(L(.)) = sum [yi ln(F(Xi’B))] + [(1-Yi) ln(1 - F(Xi’B))]

34
Q

Simplified log likelihood function form for logit model

A

ln(L(.)) = sum i=1,..,n1 Xi’B – sum i=1,..,n

ln(1 + e^Xi’B)

35
Q

How does stata maximise the log likelihood function?

A

It partially differentiates the ln(L(.)) w.r.t beta and sets = 0. We find a unique solution for beta, but we cannot write a simply algebraic expression since the function is non-linear.

36
Q

F for a PROBIT model + formula

A

F = a NORMAL distribution
F = integral between –infinity & Xi’B/sigma
(2Pi)^-0.5 exp(-Z^2/2) dZ

37
Q

What do we assume about sigma for probit model?

A

Assume sigma = 1

So we can estimate Beta and not just Beta/sigma.

38
Q

Log likelihood function for probit model

A

ln(L(.)) = sum [yi ln(Phi(Xi’B))] + [(1-yi) ln(1 - Phi(Xi’B))]

39
Q

Logit vs probit distributions

A

Logit = logistic distribution
Probit = normal distribution
- Very similar CDFs, but logistic is flatter in the tails.

40
Q

partial derivative of E(Yi) w.r.t X1 for logit

A 
dE(Yi)/dX1 = dCDF(Z)/Z x dZ/dX1
Z = B0 + B1X1i + B2X2i
dZ/dX1 = B1
41
Q

Derivative of CDF =

A

PDF

42
Q

How does the PDF differ near/away from mean?

A

Near mean: PDF (=slope of CDF) = large

At extremes: PDF = small

43
Q

What is the PDF for a logit?

A

PDF of a logistic = CDF x (1 - CDF)

PDF = e^z / (1 + e^z)^2

44
Q

Can we interpret B1?

A

NO - only a scaled version of B1

B1 x PDF = change in P(Y=1) for unit increase in X1.

45
Q

Impact of a dummy variable on CDF

A

Dummy variable = vertical displacement of CDF.

46
Q

ME of a dummy variable =

A

difference between 2 CDFs at a certain value of X1.

47
Q

How does ME of a dummy differ across distribution?

A

Near mean of X1 = larger ME

At extremes = smaller ME

48
Q

PDF of a probit model

A

phi(Z) = (2Pi)^-0.5 exp(-Z^2 / 2)

- Take CVs from standard normal table

49
Q

ME depends on i so how do we interpret?

A

Interpret at MEAN VALUES

50
Q

3 properties of MLE estimator of bj

A
  1. consistent
  2. asymptotically normal
  3. most efficient
51
Q

What test do we do for 1 restriction?

A

Approx. Z test

52
Q

What test do we do for multiple restrictions? What is DOF?

A

Chi-squared k

K = DOF = number of restrictions we are testing.

53
Q

For a multiple restriction test, what is the test statistic formula?

A

LR =2[ln(Lu) - ln(LR)]
Lu = log likelihood of unrestricted model
LR = log likelihood of restricted model

54
Q

R^2 formula for log likelihood. Why is it bad?

A

R^2 = 1 - [ln(Lw) / ln(L0)]

Where L0 = log likelihood if we only have an intercept. Bad as has no natural interpretation.

55
Q

Describe goodness of fit tests

A

Yi^ = 1 if E(yi) > 0.5 & 0 otherwise
Compare predicted and actual Y - Yi ≠ Yi^ due to Ui random shocks.
Look at proportion of total correctly predicted.

56
Q

Goodness of fit test if we adopt a simple/constant probability rule.

A

Yi^ = 1 if p > 0.5 & 0 otherwise
p = sample proportion
- We predict everyone to vote leave if the proportion voting leave > 0.5 across whole sample.
- Compare this to E(yi) one and see the gain from using the model.

METRICS TERM 1 (11 decks)

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