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2nd Year > Econometrics > Flashcards

Econometrics Flashcards

1
Q

How to formulate a model?

A
  • Statement of theory/hypothesis
  • Collect data
  • Specify mathematical model and stats theory
  • Estimate parameters
  • Check for model adequacy
  • Test hypothesis
  • Use model for predictors
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2
Q

Types of data and examples?

A

Time series: e.g GDP, unemployment. Can be both quantitative (e.g. prices) and qualitative (gender)
Cross-section: Data on variables from one point in time
Pooled: Combination of both

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3
Q

What is a linear regression?

A

Regression studying the linear relationship between dependant (explained) variables and independent (explanatory)

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4
Q

What is the population regression function?

A

Mathematical representation of the line of best fit

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5
Q

What does the error term represent?

A
  • stochastic error (random probability)
  • Represents variables not in the model
  • Randomness of human behaviour
  • Errors of measurement
  • Ockham’s razor (keep simple until proved inadequate)
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6
Q

How is a sample regression function different?

A

Used when you can only estimate values using a sample of the data. Error term is a residual (ei) and Y has a ^ as it is an estimate. Get as close to the PRF as we can

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7
Q

How does OLS work?

A

Aims to minimise the value of the residual sum of ei^2.

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8
Q

Equation for B1 and B2

A

Mean of Y - B2(Mean of X) = B1

Sum of XiYi - (n)(mean of x)(mean of y).
ALL DIVIDED BY
Sum of Xi^2 - (n)(Mean of X)^2)

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9
Q

OLS properties?

A
  • SRF passes through the sample means
  • Mean of residuals is 0
  • Sum of residuals and explanatory variables X is 0 (uncorrelated)
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10
Q

Difference between percentage increase, and percentage point increase?

A

6%- 7% = 1% percentage point increase

((7-6)/6) x 100 = 16.6% percentage increase

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11
Q

Assumptions of OLS model?

A

-Linear parameters
-X is uncorrelated with U
-E(u l xi) = 0
-Var (u) = δ^2
-No correlation of error terms (autocorrelation)
cov (ui, uj) = 0 (I and J not equal)
-No specification errors

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12
Q

What is homoscedastic variance? How is it calculated?

A

All variables have the same variance

Sum of residuals squared (RSS) / n -2 (degrees of freedom)

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13
Q

What is the Gauo- Markov theorem?

A

OLS estimators are BLUE: Best linear unbiased estimators

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14
Q

What is the central limit theorem?

A

If there is a large number of independent and identically distributed random variables then the distribution of their sum tends to a normal distribution as the sample reaches infinity

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15
Q

When is the T distribution used?

A

To test the null that Ho: B2 = 0, to see if there is a relationship between X and Y and the true variance is unknown

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16
Q

T test equation?

A

(b2 - B2) / se (b2) approx equal to t, n - (n-1)

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17
Q

Why might a one tailed test be used?

A

If you think the value is definitely above 0 for example

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18
Q

What is the total sum of squares ( sum of y^2) equal to?

A

ESS + RSS

(b2^2 x sum of (x^2)) + Sum of residuals squared

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19
Q

Proof that TSS = ESS + RSS

A 
Y = (Y^) + e
(Y - MeanY) = (Y^ - MeanY) + (Y - Y^)(e)
y = b2xi + e and Y^ = b2xi
Solve out...
Sum of y^2 = b2^2 x sum of x^2 + sum of residuals squared
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20
Q

If the line is a good fit what relationship does ESS and RSS have?

A

ESS > RSS

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21
Q

How can R^2 be calculated?

A

ESS/TSS

OR

1 - Sum of residuals / sum of y^2

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22
Q

What is a normality test and some examples?

A

Used to see if data is close to a normal distribution. Tests include histogram of residuals, probability plot and Jarque- bara test

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23
Q

How is the Jarque-Bara test conducted?

A

Test skewness and kurtosis for a normal distribution match (small value = normal)

n/6 (skew^2 + (kurt - 3)^2 /4)

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24
Q

What is forecasting?

A

Using the equation to predict the value outputted (only use numbers within the range to avoid extrapolating)

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25
Q

What does B2 measure in a multiple regression?

A

The change in the mean of Y, per unit change in X2 holding X3 constant

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26
Q

What is multicollinearity?

A

When an exact linear relationship exists between the explanatory variables

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27
Q

How to use P value for significance testing?

A

Calculate T test statistic. Calculate the P value.
“If the null hypothesis is true, what is the probability that we’d observe a more extreme test statistic in the direction of the alternative hypothesis than we did?”

Set the significance level, α (type 1 error) at 5% etc. If the P-value is less than (or equal to) α, reject the null hypothesis in favor of the alternative hypothesis.

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28
Q

What do lower P Values mean?

A

More chance of rejecting the null

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29
Q

What is the test of overall significance?

A

Ho; B2 = B3 = 0, jointly and simultaneously equal to 0, no influence on Y. Can be significant variables together even if not apart

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30
Q

Equation for the F test?

A

ESS/ D.F Variance explained by X2 and X3 over
RSS/D.F Unexplained variance

K-1 df in numerator
N-k df in the denominator

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31
Q

What does K and N represent?

A

N: Number of partial slope coefficients
K: Number of parameters (slopes and intercept)

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32
Q

What does a large F value mean?

A

More evidence that X2 and X3 do have an effect on Y

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33
Q

Equation linking F and R^2

A

F = R^2 / (K-1)
(1-R^2) (N-K)

n= Number of observations
k= number of explanatory variables
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34
Q

When R^2 = 1 what does F equal?

A

Infinity

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35
Q

TSS in terms of R^2

A

(Sum of y^2) = R^2 (Sum of y^2) + (1-R^2)(Sum of y^2)

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36
Q

What is a specification bias?

A

If X3 is ignored then X2 displays the gross effect of X2 and indirect effect of X3 by omitting the values we have a specification bias

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37
Q

Why can’t we compare R^2 values?

A

R^2 is larger the more explanatory variables there are but doesn’t account for degrees of freedom- cannot compare two values!

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38
Q

How to calculate adjusted R^2 to compare values?

A

1 - (1-R^2) ((n-1)/(n-k))

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39
Q

Properties of the adjusted R^2?

A

Adjusted R^2 is less than or equal to R^2. The more variables in the model the smaller the adjusted value compared to R^2 becomes (it can become negative)

Adjusted R^2 increases in absolute t value of the coefficient is greater than 1

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40
Q

When can we use RLS?

A

This assumes some of the variables do not belong in the model, only use this when the dependant variables are in the same form

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41
Q

F test for restricted least squares equation?

A

Fm, n-k = (R^2ur - R^2r) / m

(1-R^2ur) / (n-k)

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42
Q

What is the elasticity coefficient?

A

%change in y / % change in x

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43
Q

What does a coefficient of B2 represent on a linear y equation with a continuous variable?

y = b1 + b2x1

A

A one unit change in x generates a B2 unit change in y

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44
Q

What does a coefficient of B2 represent on a linear y equation with a log variable?

y = b1 + b2lnx1

A

A 100% change in x generates a b2 change in y

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45
Q

What does a coefficient of B2 represent on a linear y equation with a dummy variable?

y = b1 + b2D1

A

The movement of the dummy from 0 to 1 produces a B2 unit change in y

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46
Q

What does a coefficient of B2 represent on a log y equation with a continuous variable?

lny = b1 + b2x1

A

A one unit change in x generates a 100*B2 percentage change in y

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47
Q

What does a coefficient of B2 represent on a log y equation with a log variable?

lny = b1 + b2lnx1

A

A 100% change in x generates a 100*B2 percentage change in y

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48
Q

What does a coefficient of B2 represent on a log y equation with a dummy variable?

lny = b1 + b2D1

A

The movement of the dummy from 0 to 1 produces a 100*B2 percentage change in y

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49
Q

What does a coefficient of B2 represent on a dummy y equation with a continuous variable?

Dy = b1 + b2x1

A

A one unit change in x generates a 100*B2 percentage point change in the probability y occurs

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50
Q

What does a coefficient of B2 represent on a dummy y equation with a log variable?

Dy = b1 + b2lnx1

A

A 100% change in x generates a 100*B2 percentage point change in the probability y occurs

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51
Q

What does a coefficient of B2 represent on a dummy y equation with a dummy variable?

Dy = b1 + b2D1

A

The movement of the dummy from 0 to 1 produces a 100*B2 percentage point change in the probability y occurs

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52
Q

In a log linear model what does B2 represent in comparison to B3?

A

B2 measures the elasticity of Y with respect to X2, holding X3 constant (partial elasticity)

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53
Q

What is a semi-log model?

A

Used to examine growth rates, by replacing ln equations with B, for regression- Only one variable in log form. Slope coefficient measures the proportional change in Y for an absolute change in explanatory variable

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54
Q

What is a linear trend model?

A

When Y is regressed on itself (Yt). This displays the absolute changes, not the relative. (Needs a stationary error term mean and variance)

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55
Q

What are polynomial regression models?

A

When the variables are not linear but the parameters are- can still use regression analysis. Be careful for collinearity.

y= B1 + B2X + B3X^2 + B4X^3

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56
Q

How do we standardise the variables?

A

This is to reduce the effect of different units- we can subtract the mean of the variable, and divide the difference by the standard deviation

yi = Y - Mean Y / Sd

57
Q

What will the intercept equal in a standardised model?

A

0- intercept always 0 (not a regression through the origin)

58
Q

What are dummy variable models analysing?

A

The differential intercept coefficient

59
Q

Max number of dummy variables

A

M- 1 (number of dummy for each qualitative variable must be one less than the categories)
Lose 1 d.f for each one added

60
Q

What is an interaction dummy variable?

A

Joint effect of two qualitative variables

61
Q

What is a differential slope coefficient?

A

DiXi, shows how much the slope coefficient varies between the two categories

62
Q

What are coincident regressors in comparison to concurrent?

A
  • No difference in slope/intercept

- Same intercept different slopes

63
Q

What are parallel regressors in comparison to dissimilar?

A
  • Slope the same, intercepts different

- Both values different

64
Q

Problems using OLS when the dependent variable is a dummy (Y)

A
  • Binomial error term
  • Heteroscedastic variance
  • R^2 not meaningful
65
Q

Attributes of a good model?

A 
Parsimony: Occam's razor (Simple)
Identifiability: 1 estimate per parameter
Goodness of fit: R^2
Theoretical consistency
Predictive power
66
Q

Type of specification errors?

A
  • Omitting relevant variables/ including extra ones
  • Errors of measurement
  • Adopting the wrong functional form
67
Q

What problem does underfitting a model cause?

A

If the omitted variable is correlated with the included ones then the coefficients will be incorrect

68
Q

If B3 is omitted and B32 is positive what effect will it have?

A

Overestimate the significance of B2

69
Q

Problems caused by overfitting a model?

A

-OLS estimated unbiased, Variance correct, T and F valid but the estimates are inefficient with larger variances than the real model. Not BLUE, large confidence intervals

70
Q

Problems caused by errors of measurement in the dependant variables?

A

OLS unbiased but the estimated variances are larger (increases the error term)

71
Q

Problems caused by errors of measurement in the explanatory variables?

A

Biased OLS, inconsistent

72
Q

How to check for accuracy of the model specification?

A
  • Check T ratios and R^2,
  • check signs in relation to the expected theory
  • Examine residual line plot
73
Q

How to do the Mackinnon-White-Davidson test?

A
  • H0: Linear H1: Log linear. Estimate models, obtained estimated Y^
  • Obtain Z1i = ln Yi - Ln Yi^
  • Regress Y on X and Z, reject Ho if Z coefficient is significant (T test)
  • Obtain Z2i = Antilog (Lnyi^) - Y^
  • Regress ln Y on Xs or logs of X + Y, reject H1 if Z2 is significant
74
Q

How to do the RESET test? (model misspecification)

A
  • Obtain Yi, Yi^
  • Re run model adding power of Yi^ , Yi^2 etc to show relationship between residuals and estimated Y
  • Calculate F from R^2 formula
  • If F is significant we can conclude the model is misspecified
75
Q

What is the F test formula (using R^2) used in the RESET test?

A

F = (R^2new - R^2old)/ (number of new regressors)

(1-R^2new)/ (n- number of parameters in new model)

76
Q

What happens if the explanatory variables are correlated?

A

Computer refuses to compute the regression with 2 variables perfectly linearly correlated with each other, this means we cannot obtain unique estimators for all parameters

77
Q

Result of having imperfect collinearity on the estimates?

A
  • OLS still blue, although one coefficient is individual insignificant
  • Large variance, confidence intervals & standard errors
  • Insignificant T ratios due to large standard errors
  • Unstable OLS, high R^2
  • Wrong signs possible
  • Is a sample phenomena may not exist in the whole population
78
Q

Indicators of Multicollinearity?

A
  • High R^2, few significant T ratios
  • High pairwise correlations
  • Examine partial correlation (r23 with r23.4)
  • Subsidiary regressions
  • Variance inflation factor
79
Q

What do variance inflation factors show?

A

VIF measure how much the variance of the estimated regression coefficients are inflated, showing how much multicollinearity exists

80
Q

How is VIF calculated? Proof?

A

Var B2 = Standard error/ sum of X^2 (1- R2^2)

VIF = 1/1- R2^2

81
Q

Values of VIF- interpretation?

A

VIF = 1 Not correlated
1 < VIF < 5 Moderately correlated
VIF > 5 to 10 Highly correlated

82
Q

When is multicollinearity ok?

A

If the relationship is expected to continue into the future

83
Q

How to remedy multicollinearity (ideas)?

A
  • Drop a variable (theory/misspecification)
  • New data/sample (Costly)
  • Rethink the model (Take a log?)
  • Prior studies on data
  • Transform variables (can reduce issue)
  • Combine cross and time data
84
Q

What is heteroscedasticity?

A

When the error variance is nonconstant

85
Q

When is heteroscedasticity usual found?

A

In cross-sectional data (scale effect of the data)

86
Q

How to test for heteroscedasticity?

A

Sum of errors squared / n - k

87
Q

Consequences of heteroscedasticity?

A
  • No min variance, not efficient (still LU)
  • OLS biased, as variance is biased
  • T/F test and confidence intervals unreliable
88
Q

How to detect for heteroscedasticity?

A
  • Examine theory, will it be there?
  • Plot residuals (of X against Y)
  • Check for outliers
89
Q

How to do the Park test for heteroscedasticity?

A
  • Run model, square residuals, take logs
  • Run model against explanatory variables (or Y)
  • Test Ho: B2 = 0 (ln ei^2 and ln xi)
  • If not rejected then B1 has homoscedastic variance
90
Q

How to do the Glejser test?

A
  • Obtain residuals
  • Regress absolute value of e on X variable thought to be associated with heteroscedasticity
  • Ho: B2 = 0 (if rejected there is probably varying variance)
  • Needs a large sample as error term can also be heteroscedastic
91
Q

How to conduct White’s General test?

A
  • Regress, find ei
  • Run auxiliary regression with powers and (x23)
  • Obtain R^2, with no heteroscedasticity n x R^2 is approx chi squared with K-1 d.f
  • If Chi is larger than critical value then reject null of no heteroscedasticity
92
Q

When is the method of weighted least squares used?

A

When the variance of the population is known, to correct for heteroscedasticity

93
Q

Proof for Weighted least squares?

A

-Divide each bit of the equation by known variance
-Error U = Ui/ Var i
-Test Vi^2 = Vi^2 / Var i for homoscedasticity
= E (Vi^2)
= E (Ui^2/ Var i^2)
= (1/Var i^2) (E (ui^2)
= (1/Var i^2) (Var i^2) == 1

94
Q

How to use remedy heteroscedasticity when the variance in unknown?

A
  • Transform the data e.g. error variance proportional to X, so perform square root transformation
  • Trial and error to find the right deflator
95
Q

What are Whites corrected standard errors and T stats?

A

Used when heteroscedasticity is detected, White’s corrected errors take the inefficiency into account- coefficients remain unchanged, standard errors change. The tests only work asymptotically - in large samples

96
Q

What is autocorrelation?

A

Correlation between error terms, usually with time series

E(ui, uj) =/ 0

97
Q

Problem with autocorrelation?

A

OLS estimators do not work as they are not efficient

98
Q

What is inertia?

A

Sluggishness: business cycles exist in time series data, likely to be correlation

99
Q

How can model misspecification lead to autocorrelation?

A

Variable mistakes or wrong formatting of functions can lead to autocorrelation- can test by removing variables

100
Q

What is the cobweb phenomenon?

A

Supply reacts to price with a lag (1 time period) because supply decisions take time to implement

101
Q

Consequences of autocorrelation?

A
  • Not efficient, no min variance
  • Variance biased (often underestimated) and this inflates T value (F/T Unreliable)
  • R^2 unreliable
  • Inefficient variance/standard errors computed
102
Q

How to do the Durbin-Watson d test?

A

Ratio of the sum of squared differences in successive residuals to the RSS

Sum of (et - et-1)^2 / Sum of et^2

103
Q

Assumptions of the DW test?

A
  • Regression includes an intercept term
  • Non stochastic X variables
  • Ut = put-1 + Vt
  • No lagged dependant variables in use
104
Q

What is the coefficient of autocorrelation?

A

Dependence of error term on the previous value (p between -1 and 1) Known as Markov’s first order autoregressive scheme

105
Q

Link between D and P value?

A

D is approx = 2 (1-p)

P = Sum of ((et) x (et-1)) / Sum of et^2

106
Q

What do the D numbers mean?

A 
P= -1 D = 4
P= 0 D = 2
P= 1 D = 0
107
Q

How to do the Durbin Watson test?

A
  • Run regression, get residual values
  • Compute d
  • Get critical D value from the tables
  • Follow the decision rules
108
Q

What are the decision rules for the Durbin Watson test?

A

No positive A reject if 0 < d < dL
No positive A no decision if dL <= d < du
No negative A reject if 4- Dl < d < 4
No negative A reject if 4- Du < d < 4 - dL

109
Q

Major issue with the DW test?

A

The zones of indecision

110
Q

Remedial measures for autocorrelation?

A

Transform model so error term is independent

  • Write a 1 period lag (t-1 for all variables)
  • Multiply by P
  • Subtract from the original equation

(We lose 1 observation so must transform X* and Y* by rooting 1-p^2 (variable))

111
Q

What is OLS called when the remedy for autocorrelation has been applied?

A

Generalised least squares

112
Q

What is the Prais-Winsten transformation?

A

Used for small sample sizes to transform autocorrelation data after its remedied as one observation is lost

(We lose 1 observation so must transform X* and Y* by rooting 1-p^2 (variable))

113
Q

How to estimate the P used for PW transformation of autocorrelation?

A
  • P is 1, assume positive autocorrelation (no intercept)
  • Use P from DW, so P ~= 1 - d/2
  • Get P from residuals, et = pet-1 + vt (bias in small samples)
114
Q

What is the Newes- West method for autocorrelation?

A

Used in large samples- compute corrected values straight from OLS. HAC testing done on computer programmes

115
Q

What is the Runs test for autocorrelation?

A

Note the signs of the residuals, a run is uninterrupted sequence of one sign. Test for the randomness of the runs (Too many = negative , Too few = positive)

116
Q

What is the Breusch Godfrey test?

A

-Run residual test
-Now run et = normal regression + p1 Ut-1 + P2 Ut-2 etc (Regress time against original)
-Calculate NR^2
(n = number of observations in the basic series - P)
-Compare to Chi-squared tables

117
Q

What is a bilateral relationship between variables?

A

When a unidirectional relationship cannot be maintained- the Xs affect Ys, and the Y affects Xs e.g. income and consumption are linked

118
Q

Why does OLS not work on bilateral variables?

A
  • OLS is not BLUE

- Y and Ut must be correlated, so B2 is biased in small samples and inconsistent in large ones

119
Q

What is the indirect least squares method?

A

When you change the regression around to use different variables, and then solve
B1 = A1/ (1 + A2)

120
Q

Does indirect least squares work?

A

Are consistent estimators but may be slightly biased in very small samples

121
Q

What is the identification problem for simultaneous equations?

A

We can’t tell which way the relationship of the regression goes (E.g. supply or demand side). Regression only shows intersection not the slope

122
Q

When is an equation exactly identified?

A

When we can identify individual parameters, if not it is underidentified

123
Q

What is overidentification?

A

When we have more than one value for a parameter of a simultaneous equation

124
Q

What are the order conditions of identification?

A

K- number of excluded variables
M- Endogenous variables

k = m-1 Exactly
k > m-1, Over
k < m-1, Under

125
Q

What is the method of two stage least squares used for?

A

To estimate an overidentification problem

126
Q

What do we do for the method of two stage least squares?

A
  • We use a proxy for variable in place of Y so it is uncorrelated with U
  • Write into regression with U as Y
127
Q

What are dynamic models?

A

Involve change over time

128
Q

Why do lags occur?

A

Psychological: Habit, time to adapt
Tech: Consumers wait for new models/price changes
Institutional: Contracts can’t be changed continually

129
Q

Problems with adding lags to models?

A
  • No max log equation
  • Lose 1d.f for each lag
  • Multicollinearity risk
130
Q

What is the Koyck adaptive expectations technique?

A

Use Y and Yt-1 as the variables so you only lose less degrees of freedom

  • Must use Durbin H stat
  • Check for correlation with error term
  • Need larger sample
131
Q

What is a good rule of thumb to spot a “spurious” (nonsense) regression?

A

R^2 > d

132
Q

What is the unit root test?

A

-Let Yt represent stochastic time series of interest
-Ho: Yt-1 = 0 (non stationary)
-Conduct Tam test (Dickey-Fuller)
If A3 > tan we reject the null (Data is stationary)

133
Q

What is a cointegrated time series?

A

When a long run equilibrium relationship occurs between non-stationary variables (combined may be constant)

134
Q

What is the random walk theory?

A

Things like stock prices cannot be predicted on the basis of values today

135
Q

What is a drift parameter?

A

Stochastic drift is the change of the average value of a stochastic (random) process

136
Q

What is a logit model?

A

When Y is binary (0,1)

137
Q

Problems using OLS on binary models?

A
  • No guarantee OLS value will be between 0 and 1
  • Error term is binomial and heteroscedastic
  • Assumes Y increases probability with explanatory variables
138
Q

How do logit functions work?

A

ln (P/(1-P)) is used instead of Y. P represents the odds that Y equals one of the categories (e.g. p = 1 is the probability that Y=1)

139
Q

Features of the logit model?

A
  • L is linear to Y but the probabilities are not
  • Can add extra variables
  • If L (logit) is positive, when X increases the odds of Y=1 increase

2nd Year (8 decks)

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