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METRICS TERM 1 > HANDOUT 1 > Flashcards

HANDOUT 1 Flashcards

1
Q

COV(X, Y) =

A

SUM (XI - XBAR)(YI - YBAR) / N-1

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2
Q

WHY IS DOF FOR COV N-1?

A

we only need either X bar or Y bar - NOT both

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3
Q

Why is COV not that useful?

A

It is NOT scale invariant - can make it as big or small as possible by changing units

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4
Q

CORR(x,y)=

A

= COV(X,Y) / SQRT VAR(X) VAR(Y)

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5
Q

Is correlation scale invariant?

A

YES

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6
Q

What is epsilon?

A

Ei = Yi - E(Yi I Xi) = Yi - alpha - beta(Xi)

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7
Q

True but unknown relationship between X and Y. What 2 parts?

A

Yi = alpha + beta(Xi) + Ei

Systematic & random components

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8
Q

4 CLRM ASSUMPTIONS

A
  1. E(Ei I Xi) = 0
  2. V(Ei I Xi) = sigma^2
  3. COV(Ei, Ej I Xi) = 0
  4. Ei I Xi - N(0, sigma^2)
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9
Q

ei =

A

the RESIDUALS

ei = yi - y hat = yi - a - bxi

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10
Q

OLS method

A

minimise the residual sum of squares to find the line of best fit.
MIN SUM ei^2

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11
Q

2 OLS conditions

A
  1. SUM ei = 0 - zero luck on average

2. SUM ei.xi = 0 - luck independent of x

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12
Q

What does ORTHOGONAL mean?

A

Independent e.g. SUM ei.xi = 0 means ei and xi are orthogonal.

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13
Q

If sum ei = 0 is NOT satisfied, how do we change the best fit line?

A

SHIFT it up/down

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14
Q

If sum ei.xi= 0 is NOT satisfied, how do we change the best fit line?

A

ROTATE it

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15
Q

estimate of alpha = a

A

a = y bar - b xbar

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16
Q

estimate of beta = b

A

b = sum (xi - xbar)(yi - ybar) / sum (xi - x bar)^2

COV(x,y) / Var(X)

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17
Q

estimate of sigma^2 = s^2

A

s^2 = sum ei^2 / DOF = RSS / DOF

= RSS / n - 2

18
Q

properties of OLS estimators

A 
B = best - min variance of b = efficient 
L = linear - a and b are linear functions of Ei which is normal so a and b normal
U = unbiased
E = estimators
19
Q

Formula for b in terms of beta to prove unbiasedness

A 
b = beta + sum wi.ei
wi = (xi - xbar) / sum (xi - xbar)^2
E(b) = beta
20
Q

What CLRM assumption do we need for unbiasedness?

A
  1. E(Ei I Xi) = 0
21
Q

what CLRM assumptions do we need for min variance of b?

A
  1. E(Ei I Xi) = 0
  2. Var(Ei I Xi) = sigma^2
  3. COV(Ei, Ej I Xi) = 0
22
Q

Formula for variance of b conditional on x

A

V(b I X) = sigma^2 / sum(Xi - Xbar)^2

23
Q

What does V(b) depend on and how?

A
  1. Sigma^2: as variance of error term rises, variance of b rises as more uncertainty
  2. Var(X): more uncertainty in X = we have a huge data range = can pin down the slope more easily = smaller V(b)
24
Q

distribution of b

A

Normal since b = beta + sum wi.Ei so linear function of Ei which is normal

25
Q

BUT when we test b what distribution do we use and why?

A

b = normal
BUT sigma^2 unknown –> use s^2
Therefore normal –> t distribution

26
Q

Distribution of s^2 and

A

(n - 2)s^2 / sigma^2 = Chi-squared n-2

27
Q

confidence interval for beta formula

A

beta = [b +- CV.se(b)]

28
Q

does the model have to be linear in x and y?

A

NO - only needs to be linear in parameters, can be non-linear in x and y

29
Q

Interpretation of beta if linear in both x and y

A

Beta = expected change in Y for 1 unit increase in X i.e. marginal effect

30
Q

Interpretation of beta if linear Y but not X

A

If g <= 0.1: B/100 = change in Y for 1% increase in X.

If g>0.1: B x ln(1 + g) = change in Y for g
= semi-elasticity

31
Q

Interpretation of beta if non-linear Y but not X

A

If g<=0.1, Beta = expected growth rate of Y for 1 unit increase X
If g>0.1:
e^beta - 1 = expected growth rate of Y
100(e^beta - 1) = % change in Y for 1 unit rise in X = semi-elasticity

32
Q

Interpretation of beta if non-linear Y & X

A

Elasticity: dy/dx = beta
Beta = proportionate change in Y / 1 proportionate change in X
= % change in Y / 1% increase in X

33
Q

TSS identity

A

TSS = ESS + RSS

34
Q

What is TSS?

A

TSS = sum(yi - ybar)^2

total variation in the dependent variable

35
Q

What is ESS?

A

ESS = sum(yi hat - y hat bar)^2

variation in Y the model explains

36
Q

What is RSS?

A

sum ei^2

Variation in Y the model fails to explain

37
Q

Formula for R^2 and what is it?

A

R^2 = 1 - RSS/TSS = ESS/TSS

Measures the total variation in Y that the model explains. Between 0 and 1.

38
Q

We use R^2 to…

A

Compare different linear models with the SAME dependent variable

39
Q

How do we test if the model predicts new observations well?

A

hypothesis test whether en+1 is significantly different from 0 as we want E(en+1)=0

40
Q

COV(a,b)=

A

Xbar sigma^2 / sum(xi - xbar)^2

41
Q

V(en+1)=

A

sigma^2 [1 + 1/n + (xn+1 - xbar)^2/sum(xi -xbar)^2]

METRICS TERM 1 (11 decks)

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