time series, cross sectional & pooled data 1. time series data -one location’s data across...

Time Series, Cross Sectional & Pooled Data

• 1. Time Series Data

-One location’s data across time-Yearly, monthly, quarterly (every three months), weekly, daily, etc.

-ie: Canadian GDP, Enron stock value, your height, U of A tuition, world pop.


• 2. Cross-Sectional Data

-Multiple Locations at one time-Taken at same time (September report, January report, etc.)

-ie: stock portfolio, player stats, provincial GDP comparison, grade report


• 3. Pooled Data

-Combination of Time Series and Cross-sectional Data-More difficult to use-Often required due to data restrictions

General Equations

Nominal Value =(Price Index/100) X Real value

Or

Real value = Nominal value / (Price Index/100)

1.2.2 Laspeyres Price Index-uses base year quantities as weights-still = 100 in base year

Lt = ∑ pricest X quantitiesbase year

----------------------------------

∑ pricesbase year X quantitiesbase year

-tracks cost of buying a fixed (base year) basket of goods (ie: CPI)

1.2.2 Paasche Price Index-uses current year quantities as weights-still = 100 in base year

Pt = ∑ pricest X quantitiest

----------------------------------

∑ pricesbase year X quantitiest

-compares cost of current basket now to cost of current basket in base year

1.2.4 Nominal, Relative, and Real Price Indexes

Nominal Price Index –-price index for a good or service -describes movement of prices over time

ie: education, gas, coffeeNote: CPI (consumer price index) for all

goods is used to measure inflation


Relative Price Index –-price index for a good or service relative to another-describes movement of prices over time compared to another good or service

Relative Price Index = Price Index A ----------------------------------

Price Index B


Real Price Index –-price index for a good or service relative to all others-describes movement of prices over time compared to all other goods

Real Price Index = Price Index A ----------------------------------

CPI (all goods)

rreal = (1+rnom-1-inf) ----------------

(1+inf)rreal+ rreal*inf = rnom-inf (rreal*inf is small)

rreal = rnom – inf

Last example: rreal = 2%-3%=-1%

1.4.2.1 Easy Interest Formula

1.4.3.2 More Frequent Compounding

If interest is compounded m times a year, 1/m of the interest is paid each time

Modified Formula:

S = P (1+[i/m])mt

S = value after t years P = principle amounti = interest rate t = yearsm = times compounded (monthly = 12, etc)

Infinite Compounding: S = Peit

1.4.3.3 Effective Rate of InterestWhich is the better investment: 25% compounded

annually or 24% compounded monthly?

iE = effective rate of interest if

compounded annually

P (1+iE)t = P (1+[i/m])mt

Solving for iE, we get:

iE = (1+[i/m])m-1

1.4.3.4 Present Value How much do I have to invest now to have a given sum of money in the future?

PV = S/[(1+i)t]PV = present value (money invested now)S = sum needed in futurei = real, compound interest ratet = years

How does this change if it’s more than a one-time investment/payment?(ie: $100 per year for 5 years, 7% interest)PV= 100+100/1.07 + 100/1.072 + 100/1.073

+ 100/1.074

= 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7

OrPV = a[1-(1/{1+i})t] / [1- (1/{1+i})]PV = a[1-xt] / [1-x] x=1/{1+i}PV = 100[1-(1/1.07)5]/[1-1/1.07] = $438.72

1.4.3.4 Continued Deposits

1.5.1 -Stocks and Flows Summary

Type of Variable

Stock Flow

Major Characteristic

Measured at a point in time

Measured over a period (between points in time)

Examples Debts, wealth, housing, stocks, capital, tuition

Deficits, income, building starts, investment, payments

Aggregation Method

Average or

Use values from the same time each year

Sum

(Average if annualized)

1.5.2 – The User Cost of CapitalUser cost of capital = implicit rental rate

=Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )

d = depreciation – increases rental cost(more willing to rent a costly item)r = return on alternate investments

(more willing to rent given high returns)[Pkt+1 – Pkt]/Pkt = capital gains/losses

(less willing to rent a constant value item)

2.1.4 – Growth Models

The most common formulas to measure growth are:

1) [{Xt-Xt-1}/Xt-1] X 100

2) [ln(Xt)-ln(Xt-1)] X 100

3) [{dX/dt}/X] X 100

4) [dln(X)/dt] X 100

-1 and 2 work well with data

-3 and 4 require calculus

-any can be used with formulas

2.A.1.2 Rules of Derivatives

-although first principles always work, the following rules are more economical:

1)Constant Rule

If f(x)=k (k is a constant),f ‘(x) = 0

2) General Rule

If f(x) = ax+b (a and b are constants)f ‘ (x) = a


3) Power Rule

If f(x) = kxn,f ‘(x) = nkxn-1

4) Addition Rule

If f(x) = g(x) + h(x),f ‘(x) = g’(x) + h’(x)


5) Product Rule

If f(x) =g(x)h(x),f ‘(x) = g’(x)h(x) + h’(x)g(x)-order doesn’t matter

6) Quotient Rule

If f(x) =g(x)/h(x),f ‘(x) = {g’(x)h(x)-h’(x)g(x)}/{h(x)2}-order matters-derived from product rule


5) Product Rule

If f(x) =(12x+6)x3

f ‘(x) = 12x3 + (12x+6)3x2

= 48x3 + 18x2

6) Quotient Rule

If f(x) =(12x+1)/x2

f ‘(x) = {12x2 – (12x+1)2x}/x4

= [-12x2-2x]/x4

= [-12x-2]/x3


7) Power Function Rule

If f(x) = [g(x)]n,f ‘(x) = n[g(x)]n-1g’(x)-special case of the chain rule

8) Chain Rule

If f(x) = f(g(x)), let y=f(u) and u=g(x), thendy/dx = dy/du X du/dx

2.A – More Derivatives

1) Natural Logs

If y=ln(x),

y’ = 1/x

-chain rule may apply

If y=ln(x2)

y’ = (1/x2)2x = 2/x

2.A – More Derivatives

2) Trig. Functions

If y = sin (x),

y’ = cos(x)

If y = cos(x)

y’ = -sin(x)

-Use graphs as reminders

2.2 Mathematical Models of Economic Relationships

Consumption Function – slope = mpc

Consumption = 100+0.5income

Mpc = dc/di = 0.5

Consumption = 100+0.5income-0.02income2

Mpc = dc/di = 0.5-0.04income

Are any other functional forms viable for consumption?

In this case, we see the slopeincreasing as tincreases, transitioningfrom a negativeslope to a positiveslope.

A second derivative would discover this fact, aid in the sketching of the graph, and confirm a minimum point on the graph. Here x’’ is positive.

2.A Second Derivativesx=15-10t+t*t

-15

-10

-5

0

5

10

1 2 3 4 5 6 7 8

t

x x

2.A – Implicit Differentiation Rules

1) Take the derivative of EACH term on both sides.

2) Differentiate y as you would x, except that every time you differentiate y, multiply that term by dy/dx (or y’)

Ie: 14=7x+9x2-yd(14)/dx=d(7x)/dx+d(9x2)/dx-dy/dx0 = 7 + 18x – y’y’=7+18x

2.A.1.3 Derivative Applications - Graphs

Graphing Steps:

i) Evaluate f(x) at x=0, ∞, - ∞, or a variety of values

ii) Determine where f(x)=0

iii) Calculate slope - f ’(x) - and determine where it is positive and negative

iv) Identify possible maximum and minimum co-ordinates where f ‘(x)=0. (Don’t just find the x values)

2.A.1.3 Derivative Applications - Graphs

Graphing Steps:

v) Calculate the second derivative – f ‘’(x) and use it to determine max/min in iv

vi) Using the second derivative, determine the curvature (concave or convex) at other points

vii) Check for inflection points where f ‘’(x)=0

2.2.2 Elasticities-to avoid this problem, economists often utilize ELASTICITIES

-elasticities deal with PERCENTAGES and are therefore more useful across a variety of ranges

ELASTICITY = a PROPORTIONAL change in y from a PROPORTIONAL change in x

Example: elasticity of demand:

η = Δy/y / Δx/x

= (Δy/Δx) (x/y)

= (dy/dx) (x/y)

2.2.2 Elastic LogsThe MAIN reason to use logs in economic formulae is to more easily calculate elasticities:

E = dy/dx * y/x= (1/y) dy/dx (x)= (dlny/dx) dy/dx (dx/dlnx)= (dlny/dlnx) dx/dx (dy/dy)= (dlny/dlnx)

2.3 Interpreting Parameters

Unfortunately for economists, our employers are not awed and amazed by elegant equations – they want to know what the elegant equations mean.

Intercepts, slopes, curvature and elasticity are thus far tools to explain models.

Parameter explanation is what employers want.

2.3.1 Simple Example

Let mark = 60 + 4 studyMark = percentage mark on midtermStudy = hours of study (up to 10 – it’s the

night before)Parameter Explanation:60 = intercept – without studying, you’d get

a 60% on the exam, you genius you4 = coefficient of study – every extra hour

spent studying increases your mark by 4%

2.A The Partial DerivativeIt is often impossible to analyze all various

movements of explanatory variables and their impact on the dependent variable.

Instead, we analyze one variable’s impact, assuming ALL OTHER VARIABLES REMAIN CONSTANT

We do this through the partial derivative.

2.4 The error termAlthough economists try to model real behavior,

their attempts are not always 100% accurate, for a variety of reasons:

1) Excluded variables

2) Random events (shocks)

3) Error in data collection

4) Economist stayed up late watching the hockey game (which could cause the above)

2.A OptimizingThere are three steps for optimization:1) Find where f’(x)=0. This is your FIRST ORDER CONDITION (FOC) and gives

potential maxima/minima.

2) Evaluate f’’(x) at your potential maxima/minima. This is your SECOND ORDER CONDITION (SOC) and determines maxima/minima/inflection point status

3) Obtain the co-ordinates of your maxima/minima

2.A Local vrs. Global

Thus far, our efforts have revealed LOCAL maxima and minima.

It is possible, however, that such these values are not the maximum or minimum possible.

These values may not be the best policy decision for a government, individual, or firm.

3.2 Probabilities

• Probabilities are assigned to the various outcomes of random variables

Terminology:Sample Space – set of all possible outcomes

from a random experiment-ie S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}-ie E = {Pass exam, Fail exam, Fail horribly}

Event – a subset of the sample space-ie B = {3, 6, 9, 12} ε S-ie F = {Fail exam, Fail horribly} ε E

3.2 Probabilities

Terminology:Mutually Exclusive Events – Two events are

mutually exclusive if they cannot occur at the same time-ie: rolling both a 3 and an 11; being both dead and alive; having both a son and a daughter (and only one child)

Exhaustive Events – cover all possible outcomes-ie: a dice roll must lie within S ε [2,12]-ie: a person is either married or not married

3.2 Probability

Probability = measure of likelihood of an event occurring (between 0 and 1)

P(a) = Prob(a) = probability event a will occurProb (Y=y) = probability that the random

variable Y will take on value yIf Prob(a) = 0, the event will certainly never

occur (ie: your instructor turns into a giant llama)

If Prob(b) = 1, the event will certainly occur (ie: the sun will rise tomorrow)

3.2 Probability Rules

1) P(a) must be greater than or equal to 0 and less than or equal to 1 : 0≤P(a) ≤1

2) If a set of events {A,B,C} are exhaustive, then

P(A or B or C) = 1Ie: Prob. a die roll is between 2 and 123) If a set of events {A,B,C} are mutually

exclusive, then P(A or B or C)=P(A)+P(B)+P(C)

Ie: Prob. of drawing a heart or spade

3.3 Expected Values

Expected Value – measure of central tendency; center of the distribution; population mean

Discrete Variable:E(Y) = Σyf(y)

Continuous Variable:E(Y) = ∫f(y)dy

3.3.1 Properties of Expected Values

a) Constant PropertyE(a) = a if a is a constant or non-random

variableIe: E(14)=14Ie: E(b1+b2Xi) = b1+b2Xi

b) Constants and non-random variablesE(a+bW) = a+bE(W)If a and b are non-random and W is random

3.4 Variance Formula

Var(Y) = E(Y-E(Y))2

= E(Y2) – [E(Y)]2

Discrete Random Variable:Var(Y) = Σ(y-E(Y))2f(y)

Continuous Random Variable:Var(Y) = ∫(y-E(Y))2f(y)dy

3.4.1 Properties of Variance

a) Constant PropertyVar(a) = 0 if a is a constant or non-random

variableIe: Var(14)=0Ie: Var(b1+b2Xi) = 0

b) Constants and non-random variablesVar(a+bW) = b2 Var(W)If a and b are non-random and W is random

3.4.1 Properties of Variance

c) Covariance PropertyIf W and V are random variables, and a, b,

and c are non-random, then

Var(a+bW+cV) = Var(bW+cV)= b2 Var(W) + c2 Var (V)

+2bcCov(W,V)Where Covariance will be examined in 3.7

3.5 Common Economic DistributionsIn order to test assumptions and models,

economists need be familiar with the following distributions:

Normal t Chi-square FExamples and explanations of these tables

are available at http://www.statsoftinc.com/textbook/sttable.html

http://www.statsoftinc.com/textbook/sttable.html

http://www.statsoftinc.com/textbook/sttable.html

3.5 Normal Distribution

The Normal (Z) Distribution produces a bell curve with a mean of zero and a standard deviation of one.

The probability that z>0 is always 0.5 The probability that z<0 is always 0.5 Z-tables generally measure area from the

centre Probabilities decrease as you move from the

mean of zero

3.5 Converting to a normal distributionZ distributions assume that the mean is

zero and the standard deviation is one.If this is not the case, the distribution

needs to be converted to a normal distribution using the following formula:

Z = (x-u)/sdWhere x = value

u = meansd= standard deviation

3.5 t-distribution

t-distributions can be 1-tail or 2-tail testsInterpolation is often needed within the

tableExample: Find the critical t-value (t*) that cuts of 1%

of the right tail with 35dfFor 1T=0.01, df 30 gives t*=2.457

df 40 gives t*=2.423A good approximation of df 35 would be:t*=(2.457+2.423)/2 = 2.440

3.5 chi-square distribution

Chi-square distributions are 1-tail testsInterpolation is often needed within the

tableExample: Find the critical chi-squared value that cuts

of 5% of the right tail with 2dfFor Right Tail = 0.05, df=2Critical Chi-Squared Value = 5.99146

3.5 F-distribution

F-distributions are 1-tail testsInterpolation is often needed within the

tableExample: Find the critical F value (F*) that cuts of 1%

of the right tail with 3df in the numerator and 80df in the denominator

For Right Tail = 0.01, df1=3, df2=80,df2=60 gives F*=4.13 df2=120 gives

F*=3.95

3.5 Interpolation

df2=60 gives F*=4.13 df2=120 gives F*=3.95

Since 80 is 1/3rd of the way between 60 and 120:

60 80 100 120Our F-value should be 1/3 of the way between

4.13 and 3.95:4.13 ? 3.95Approximatation:

F*=4.13-(4.13-3.95)/3=4.07

3.6 Joint Probability Density Functions

Joint Probability Density Function--summarizes the probabilities

associated with the outcomes of pairs of random variables

f(p,q) = Prob(P=p and Q=q)∑ f(p,q) = 1

Similar statements are valid for continuous random variables.

3.6 Joint and Marginal Pdf’s

Marginal (individual) pdf’s can be determined from joint pdf’s. Simply add all of the joint probabilities containing the desired outcome of one of the variables.

Ie: f(Y=7)=∑f(Y=7,Z=zi)

Probability of Y=7 = sum of ALL joint probabilities where Y=7

3.6 Conditional Probability Density Functions

Conditional Probability Density Function--summarizes the probabilities

associated with the possible outcomes of one random variable conditional on the occurrence of a specific value of another random variable

Conditional pdf = joint pdf/marginal pdfOrProb(a|b) = Prob(a&b) / Prob(b)(Probability of “a” GIVEN “b”)

3.6 Statistical Independence

If two random variables (W and V) are statistically independent (one’s outcome doesn’t affect the other at all), then

f(w,v)=f(w)f(v)Therefore:

1) f(w)=f(w|any v)2) f(v)=f(v|any w)

As seen in the previous example.

3.6 Conditional Expectations and VarianceAssuming that our variables take numerical values (or can be interpreted numerically), conditional expectations and variances can be taken:

E(P|Q=500)=Σpf(p|Q=500)Var(P|Q=500)=Σ[p-E(P|Q=500)]2f(p|Q=500)

Ie) money spent on a car and resulting utility (both random variables).

3.7 Discrete and Continuous Covariance

Discrete Random Variable:

Continuous Random Variable:

v w

wvfwEwvEvWVCov ),()())(((),(

v w

wvwvfwEwvEvWVCov ),()())(((),(

3.7 Correlation

Correlation:

)()(),(),( WsdVsd

WVCovWVCor

)()(

),()())(((),(

wVarvVar

wvfwEwvEvWVCor v w

3.8 EstimatorsPopulation Expected Value:

μ = E(Y) = Σ y f(y)

Sample Mean:

__Note: From this point on, Y may be expressed as Ybar (or any other variable - ie:Xbar). For example, via email no equation editor is available, so answers will be in this format.

N

YY i

3.8 Estimators

Population Variance:

σ2 = Var(Y) = Σ [y-E(y)] f(y)

Sample Variance:

1

)( 22

N

YYS iy

3.8 Estimators

Population Standard Deviation:

σ = (σ2)1/2

Sample Variance:

Sy = (Sy2)1/2

3.8 Estimators

Population Covariance:

Cov(V,W)=∑∑(v-E(v))(w-E(w))f(v,w)

Sample Covariance:

1

))((),(

N

WWVVWVCov ii

3.8 Estimators

Population Correlation:

ρvw = corr(V,W)= Cov(V,W)/ σv σw

Sample Correlation:

rvw = corr(V,W)= Cov(V,W)/ Sv Sw

3.8 Estimators

Population Regression Function:

Yi = b1 + b2Xi + єiEstimated Regression Function:

ii XbbY 21

ˆˆ

3.8 Estimators

OLS Estimation:

B2hat = ∑(Xi-Xbar)(Yi-Ybar)

---------------------- ∑(Xi-Xbar)2

B1hat = Ybar – B2hatXbar ^

Note: b2 may be expressed as b2hat

XbYb

XX

YYXXb

i

ii

21

22

ˆˆ

)(

))((ˆ

3.9.2 Fitted or Predicted Values

From the above we see that often the actual data points lie above or below the estimated line.

Points on the line give us ESTIMATED y values for each given x.

The predicted or fitted y values are found using our x data and our estimated b’s:

ii XbbY 21ˆˆ

3.9.3 Estimating Errors or Residuals

The estimated y values (yhat) are rarely equal to their actual values (y).

The difference is the error term:

YYE iiˆ

3.9.5 Statistical Properties of OLS

In our model:Y, the dependent variable, is made up of

two components:

a) b1 + b2Xi – a non-random component that indicates the effect of X on Y. In this course, X is non-random.

b) Єi – a random error term representing other influences on Y.


Error Assumptions:a) E(єi) = 0; we expect no error; we

assume the model is completeb) Var(єi) = σ2; the error term has a

constant variancec) Cov(єi, єj) = 0; error terms from two

different observations are uncorrelated. If the last error was positive, the next error need not be negative.


OLS Estimators are Random Variables:a) Y depends on є and is thus random.b) B1hat and B2hat depend on Yc) Therefore they are randomd) All random variables have probability

distributions, expected values, and variances

e) These characteristics give rise to certain OLS estimator properties.

3.9.5 OLS is BLUE

We use Ordinary Least Squares estimation because, given certain assumptions, it is BLUE:

B est

L inear

U nbiased

E stimator

3.10.1 Formula

Given

We have an upper limit of:

And a lower limit of:

Or:

1}**{ stXstXP

stX *

stX *

stXCI *

3.11 Hypothesis Testing

Testing Consistency of a Hypothesized Parameter:

1) Form a null and an alternate hypothesis.H0 = null hypothesis = variable is equal to a

numberHa = alternate hypothesis = variable is not

equal to a numberIe)H0: b2=0

Ha: b2≠0



2) Collect appropriate sample data3) Select an acceptable probability (α) of rejecting

a null hypothesis when it is true-Type one error-Lower α, more unlikely to find a sample that

rejects the null hypothesis- α is often 10%, 5%, or 1%



4) Construct an appropriate test statistic-ensure the test statistic can be calculated from

the sample data-ensure its distribution is appropriate to that being

tested (ie: t-statistic for test for mean)



5) Establish (do not) reject regions-Construct bell curve

-Tails are Reject H0 regions

-Centre is Do not Reject H0 regions



6) Compare the test statistic to the critical statistic-If the test statistic lies in the tails, reject-If the test statistic doesn’t lie in the tails, do not

reject-Never Accept

7) Interpret Results

4.1.2 Measuring Goodness of FitOn average, OLS works well: The average of the estimated errors is zero The average of the estimated Y’s is always the average of the

observed Y’s

Proof:

4.1.2 Measuring Goodness of FitR2 is constructed by dividing the variation of Y

into two parts:

1) Variation in fitted Yhat terms. This is explained by the model

2) Variation in the estimated errors. This is NOT explained by the model.

4.2 Hypothesis TestingSo far, we have assumed: The error term, єi, is random with

E(єi)=0; no expected error

Var(єi)=σ2; constant variance Cov(єi, єj)=0; no covariance between

errors

Now we add the assumption that the error term is normally distributed. Therefore:

Єi ~ N(0,σ2)

4.2 Hypothesis TestingIf the error is normally distributed, so will be the Y

term (since the randomness of Y depends on the randomness of the error term). Therefore:

E(Yi) = E(b1+b2Xi+єi)=b1+b2Xi

Var(Yi) = Var(b1+b2Xi+єi)=Var(єi) = σ2

(Given that only Y and є are random, plus our error term assumptions.)

Therefore:

Yi ~ N(b1+b2Xi, σ2)


Since we don’t know σ2, we can estimate it:

This gives us estimates of the variance of our coefficients:

4.3.1 Deriving a Confidence Interval

Step 1: Recall Distribution

We know that:

(b1hat-b1)/se(b1hat) has a t distribution with N-2 degrees of freedom

(b2hat-b2)/se(b2hat) has a t distribution with N-2 degrees of freedom

This was derived under hypothesis testing using central limit theorems.

4.3.1 Deriving a Confidence IntervalStep 4: Rearrange for CI:

Thus the 100(1-α)% Confidence Interval is defined by the range:

By repeatedly calculating Confidence Intervals using OLS, 100(1- α)% of these CI’s will contain the true value of the parameter (b1).

4.4 Prediction in Simple Regression ModelsLinear Model:

YPred=b1hat +b2hatX* +0

b1hat estimates b1

b2hat estimates b2

0 estimates the error term

Model evaluated at X*

4.4 Prediction in Simple Regression ModelsSolution:

Since we can estimate σ2/2,

QPred =exp{g1hat + g2hat ln(P*) + σhat2/2}

g1hat estimates b1

g2hat estimates b2

σhat2 estimates σ2

Model evaluated at P*

time series, cross sectional & pooled data 1. time series data -one location’s data across...

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