3A Demand Estimationl

7/31/2019 3A Demand Estimationl

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Demand Estimation

&

Forecasting


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Definition of Elasticity of demand

Price Elasticity of demand:

Income Elasticity:

Cross Price Elasticity:

q

p

p

qep

q

I

I

qeI

q

p

p

qe r

r

pr


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Interpreting the Price Elasticity ofDemand: How Elastic Is Elastic?

Demand is elastic if the price elasticity ofdemand is greater than 1

Inelastic if the price elasticity of demand isless than 1, and

Unit-elastic if the price elasticity ofdemand is exactly 1.


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Highway departmentcharges for crossing a

bridge


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Nature of goods according to Income

elasticity

eI >0 => Normal Goods

eI < 0 => Inferior Goods

eI Necessities

eI >1 => Luxury Goods


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Cross-Price Elasticity

Goods are substitutes when the cross-priceelasticity of demandis positive

e.g. Coke & Pepsi, Zen & Santro

Goods are complements when the cross-priceelasticity of demand is negative

e.g. tea & sugar, petrol & petrol-driven car


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Alcoholic Beverages elasticities (e)

Many public policy issues are related to the

consumption of alcoholic beverages

Spirits refer to all beverages that contain

alcohol other than beer & wine

Price elasticity (epb ) of dd for beer -0.23

Cross-price (epb,pw) 0.31

Cross-price (epb,ps) 0.15

Income elasticity (eIb) -0.09

Income elasticity (eIw) 5.03

Income elasticity (eIs) 1.21


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Alcoholic Beverages elasticities (e)

Demand for beer inelastic

10% increase in beer price will result in 2.3% decrease in

beer demand

Wine & spirit are substitutes for beer

A 10% increase in wine price will result in 3.1% increase inthe quantity of beer demanded

Similarly for spirit, a 10% increase will increase 1.5%

increase in quantity of beer demand

Beer is an inferior good 10% increase in income will result in 0.9% decline in

quantity of beer demanded

Both wine & spirit are luxury goods as income

elasticities are >1


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Determinants of Demand

Consumer Income (more purchasing power) Price of the product

The prices of related goods

Substitute Goods (e.g. petrol vs. diesel)

Complementary Goods (diesel car & diesel sale)

Consumer expectations of future price & income

Population & growth of the economy

Consumer tastes and preferences

Demand=f(Y, Pr, Po, ..)


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Methods of Demand Estimation


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Interview and Experimental Methods

Expert Opinion

Consumer Interviews/ surveys

Interviews can solicit useful information when market data

is scarce.

Sample selection to represent consumer population & skillof surveyors are important

Market Experiments

Controlled experiments in test markets can generate usefulinsights

Advantage over surveys as it reflect actual consumer

behavior

Experiments can be expensive


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General Empirical Demand Specification

Q = f(P, M, PR N)

Where,

Q= quantity demanded

P = Price of the good

M = Consumers income

PR = Price(s) of the related product(s)

N = Number of buyers

Linear form of the demand function is

Q = a + bP + cM + dPR + eNWe need to know the value of a, b, e..how ??

There are many ways but most common one isthrough Regression Analysis


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Regression Analysis

Regression analysis is concerned withthe study of the relationship betweenone variable called explained ordependent variable (y) and one or moreother variables called independent orexplanatory variables (x1, x2xn)

Y = f (x1, x2xn)


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Methodology for Regression Analysis

Theory

Mathematical model of theory

Econometric model of theory

Data collection

Estimation of econometric model

Hypothesis testing

Forecasting

Using the model for policy purpose


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Specification of Mathematical & Econometric Model

Y = B1 + B2X; Mathematical model (Deterministic)

Y = B1 + B2X + u

Econometric model (Example oflinear regression model)Y Dependent Variable; X Independent Variable; u Error term

B1 & B2 are parameters to be estimated

X

Y

B2* * *

* * *

X

Y

B1


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Econometric Model

Actual = systemic part+ random error Say, Consumption (C) = Function (f) of income

(I) with error (u)

C = f(I) + u

u represents the combined influence ondependent variable of a large number ofindependent variables that are not explicitly

introduced in the regression model We hope that influence of those omitted or

neglected variables is small and at bestrandom


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Assumptions

The relationship between X & Y is linear

The Xs are non-stochastic variables whosevalues are fixed The error has zero expected value; E(u)=0 The error term has constant variance; E(u2) = 2

homoscedastic Errors are statistically independent.

Thus, E(ui uj)=0 for all i j no autocorrelation

The error term is normally distributed;

u ~ N (0, 2) uiXi = 0 u & X are uncorrelated Y~ N (B1 + B2X, 2 )


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Linearity Assumption

The term linearin a simple regression model does not mean a

linear relationship between variables, but a model in which theparameters enter the model in a linear way

A function is said to be linear in parameter if itappears with a power of one and is not multiplied ordivided by any other parameters


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Useful Functional Form

Linear:

Reciprocal

Log-Log


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Useful Functional Form

Log-linear

Linear-log

Log-inverse


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Population Regression Function

Let Y represents weekly expenditure onlottery &

X represents weekly personal disposable

income

For simplicity, we assume a hypothetical

population of 100 players, which has beendivided into 10 PDI classes in incrementsof $25 starting with $150 and endingwith $375


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Weekly exp on Lotto and weekly PDI

150 175 200PDI,X

Y, Weekly exp on LottoPRL

E(Y/Xi) = B1 + B2X

(mathematical)

Yi = B1+ B2Xi+ui(stochastic,

individual values

different from mean

values)

B1 B2 parameters

225

uiui


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PRF

For any X value, there are 10 Y values

Also, there is a general tendency for Yto increase as X increases people with

higher PDI likely to spend more onlottery.

This will be more clear if we take mean

value of Y corresponding to various Xs If we connect various mean values of Y,

the resulting line is called PRL


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SRF

Here, SRL: =b1+b2Xi Where , b1,b2 are estimator of E(Y/Xi), B1 and B2

An estimator is a formula that suggests how

we can estimate population parameter

A particular numerical value obtained by theestimator in an application is an estimate

Stochastic SRF: Yi=b1+b2Xi+ei, ei=estimator ofui


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SRF

Thus, ei = Yi Granted that SRF is onlyapproximation of PRF, can we find amethod that will make thisapproximation as close as possible?

Or, how should we construct SRF sothat b1 & b2 are as close as B1 & B2?


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Population & Sample Regression Line

Suppose we would like to estimate demand ofrice in Gurgaon and the demand =f(income)

One way to estimate this is to go each person

in Gurgaon to collect data on income and rice

consumption to estimate the equation

C = B1 + B2 M, where B1 & B2 are parameters

to be estimated

Other way is to collect data from a sample of

say 100 people and estimate C = b1 + b2 M


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Population & Sample Regression Line

However, for another sample we may get C =c1 + c2 M and so on

We cannot say which SRL represent PRL

Can we estimate PRF from sample data? Granted that SRF is only approximation of PRF,

can we find a method that will make this

approximation as close as possible? Or, how should we construct SRF so that b1 &

b2 are as close as B1 & B2?


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Estimation of parameters:

Method of Ordinary Least Squares

We have, ei = Yi = Yi - b1 - b2Xi

Objective is to choose b1 & b2 so that ei are assmall as possible

OLS states that b1 & b2 should be chosen in such away that RSS in minimum

Thus, minimise ei2= (Yi - b1 - b2Xi)2

b2= xiyi/ xi2 =

b1 = - b2

(t

X - X

_

) (t

Y - Y

_

)/(t

X - X

_

)2


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Estimating coefficients

Consider a firm with a fixed capital stock that has

been rented under a long-term lease for Rs 100 perproduction period. Other input of the firmsproduction process is labor, which can be increased ordecreased depending on the firms needs. So, cost ofthe capital input is fixed and cost of labor is variable.

The manager of the firm wants to know therelationship between output and cost. This will allowthe manager to predict the cost of any specified rateof output for the next production period

The manager is interested to estimate thecoefficients b1 and b2 of the function

Y = b1 + b2 X, where Y is total cost and Xis total output


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Estimates

Cost(Yt)

Output(Xt) t

Y - Y_

t

X - X_

(t

X - X_

)2

(t

X - X_

) (t

Y - Y_

)

100 0 -137 -12.29 151.04 1645.45

150 5 -87.14 -7.29 53.14 635.25

160 8 -77.14 -4.29 18.4 330.93

240 10 -2.86 -2.29 5.24 -6.55

230 15 -7.14 2.71 7.34 -19.35

370 23 132.86 10.71 114.7 1422.93

410 25 172.86 12.71 161.54 2197.05

Y_

=

237.14

X_

=

12.29

(t

X - X_

)2

= 511.4

(t

X - X_

) (t

Y - Y_

)

=6245.71


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Estimates

Y = 87.08 + 12.21 X

One unit change in X results in 12.21 units change in Y

b2 = ( tX - X) ( tY- Y )/( tX - X)2

= 12.21

b1 = Y - b2 X= 237.1412.21 (12.29) = 87.08

EVIEWS


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Estimates

So far we have estimated b1 & b2 using OLS

It is evident that least square estimates area function of sample data

Since the data are likely to change fromsample to sample, the estimates will alsochange

Therefore, what is needed is some measure ofreliability or precision of the estimators b1 &b2, which can be measured by standard error


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Variances (& SEs) of OLS estimators

(T-2) is called dof, number of independent observations, as we loose 2 dof

to compute b1 & b2 in estimating Y(cap)


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Computing sources of variation

YtTotalVariation

(t

Y - Y )2

tY = 1

b +2

b XtExplainedVariation

(t

Y - Y )2

UnexplainedVariation

(t

Y -t

Y )2

100 18,807.38 87.08 22,518 166.93

150 7593.38 148.13 7922.78 3.5160 5950.58 148.76 2743.66 613.06

240 8.18 209.18 781.76 949.87

230 50.98 270.23 1094.95 1618.45

370 17,651.78 357.91 17,100.79 4.37

410 29,880.58 392.33 24,083.94 312.23Y = 237.14 (

tY - Y )

2

=79,942.86

(t

Y - Y )2

=76,245.88

(t

Y -t

Y )2

=3668.41


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Standard error of estimate

Var (b2) = [ ( tY -t

Y )2/(T2)]/(

tX - X )

2

= [3668.41/ (7 -2)]/511.4 = 1.4161

se (b2) = 4161.1 = 1.19

= 87.08 + 12.21 X

(***) (1.19)

where figures in parentheses are estimated std. errors, which measuresvariability of estimates from sample to sample

t-test is used to determine if there is a significant relationship betweendependent variable and each independent variable

The test requires that s.e. of the estimated regression coefficient be computed


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Hypothesis testing

Say, prior knowledge or expert opinion tells us that trueaverage price to earning (p/e) ratio in the population ofBSC is 20

Suppose a particular random sample of 30 stocks givesthis estimate as 23

Is the value of 23 statistically differentfrom 20?

Due to sample fluctuations it is possible that 23 may notstatistically different from 20

In this case we may not reject the hypothesis that truevalue of p/e is 20

This can be done through hypothesis testing

h i i


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Hypothesis testing

Suppose someone suggests that X has no effect

in Y Null hypothesis: H0: B2 = 0

If H0 is accepted, there is no point in including Xin the model

If X really belongs to the model then one wouldexpect that H0 must be rejected againstalternate hypothesis H1, which says thatB2 0

It could be positive or negative

Though in our analysis b2 0, we should not lookat numerical results alone because of samplingfluctuations


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Statistical evaluation of regression results

This can be done through ttest t-test: test of statistical significance of each

estimated regression coefficient

b: estimated coefficient

SEb: standard error of the estimated coefficient

Rule of 2: if absolute value of t is greater than 2,estimated coefficient is significant at the 5% level

If coefficient passes t-test, the variable has a trueimpact on demand

bSE

b

t


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CI Vs TOS

In CI approach, we specify a plausible range

of values for the true parameter and find outif CI includes the hypothesized value of theparameter

If it does, we do not reject Ho but if it lies

outside CI, we can reject Ho In test of significance approach, instead of

specifying a range of values, we pick a specificvalue of the parameter suggested by Ho

In practice, whether we use CI approach orTOS approach of hypothesis testing is amatter of personal choice and convenience


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Test of significance

One property of normal distribution is thatany linear function of normally distributedvariables is itself normally distributed

Since b1 and b2 are linear function of u, whichis normally distributed

Therefore, b1 and b2 should also be normallydistributed


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Test of significance

b1 ~ N (B1,2

1b )

b2 ~N (B2,2

2b )

Z = (b2 B2)/ se(b2) = (b2 B2)/ / ( 2ix ) ~ N ( o . 1 )

Where xi = (Xi - X )

Since we dont know , w e h a v e t o u s e t h e e s t i m a t e o f .

In that case, (b2 B2)/ / ( 2ix ) ~ tn-2

= estimator (b2) hypothesized value (B2*)/se of estimator (b2)

If the absolute value of this ratio is equal to greater than

the table value of t for (n-2) dof, b2 is said to bestatistically significant

In our case, t = b2/ [se(b2)] = 12.21/1.19 = 10.26 > tablevalue of t stat at 95% confidence interval and at 5 dof,which is 2.015

So H0 : B2 = 0 is rejected

hypothetical distribution under


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b2 b2+sd b2+2.58sdb2-sdb2-2.58sd

0.5%0.5%

5% level

1% level

hypothetical distribution under

0220 :H

acceptance region for b2

00000

5

b2

t-statistic

The diagram show the acceptance region and therejection regions for a 5% and 1% significance

test.

2.5%2.5%


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Explanatory power of a model

Y X

Y X

Y X


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Breakdown of total variation

X

Xt

(Xt,Yt)

SRF

Total Variation

(Yt - )

(t - )=variation in Yt explained

by regression

et=(Yt- t)


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Decomposition of Sum of Squares

(Yt - ) = (t - ) + (Yt -t)

After squaring both sides and algebraicmanipulations, we get

TSS = ESS + RSS

2 2 2 ( ) ( ) ( )t t tY Y Y Y Y Y

2

2

2

( )

( )t

Y YExplained VariationR

Total Variation Y Y

G d f fi 2


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Goodness of fit: R2

Value of R

2

ranges from 0 to 1 If the regression equation explains none of

the variation of Yi (i.e. no relationshipbetween X & Y), R2 will be zero

If the equation explains all the variation, R2will be one

In general, higher the R2 value, the better the

regression equation A low R2 would be indicative of a rather poor

fit

2V Ex


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Three Variable Regression

Model


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Yi = B1+B2X2i+B3X3i_ Nonstochastic form,PRF

Yi = B1+B2X2i+B3X3i+ui stochastic

B2, B3 called partial regression or partial

slope coefficients

B2 measures the change in mean value of Y,per unit change in X2 holding the value of

X3 constant Yi = b1+b2X2i+b3X3i+ei SRF


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Assumptions

Linear relationship

Xs are non-stochastic variables.

No linear relationship exists between two or

more independent variables (no multi-collinearaity). Ex:X2i = 3 +2X3

Error has zero expected value, constantvariance and normally distributed

RSS = e2 = (Yii)2= (Yi b1-b2X2i-b3X3i)2


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Testing of hypothesis, t-test

Say, i = -1336.09 + 12.7413X2i+85.7640X3i

(175.2725) (0.9123) (8.8019)

p=0.000 0.000 0.000

R2 = 0.89, n =32

H0: B1=0, b1/se(b1)~ t(n-3)

H0: B2=0, b2/se(b2)~ t(n-3)

H0: B3=, (b3 - )/se(b3)~ t(n-3)


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Testing Joint Hypothesis, F Test

H0

: B2

= B3

= 0Or, H0 : R

2= 0

X2 & X3 explain zero percent of thevariation of Y

H1: At least one B 0

A test of either hypothesis is called a test

of overall significance of the estimatedmultiple regression

We know, TSS = ESS + RSS


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F test

If computed F value exceeds critical F value, we

reject the null hypothesis that the impact ofexplanatory variables is simultaneously equal to zero

Otherwise we cannot reject the null hypothesis

It may happen that not all the explanatoryvariables individually have much impact on dependentvariable (i.e., some of the t values may bestatically insignificant) yet all of them collectivelyinfluence dependent variable (H0 is rejected in Ftest)

This happen only we have the problem ofmulticollinearity

f


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Specification error

In this example we have seen thatboth the explanatory variables areindividually and collectively differentfrom zero

If we omit any one of theseexplanatory variable from our model,then there would be specification

error

What would be b1, b2 & R2 in 2-

variable model?

f


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Specification error

i = -1336.09 + 12.7413X2i+85.7640X3i(175.2725) (0.9123) (8.8019)

p=0.000 0.000 0.000

R2 = 0.89, n =32

i = -191.66 + 10.48X2(264.43) (1.79)

R2 = 0.53

i = 807.95 + 54.57X3i(231.95) (23.57)

R2 = 0.15


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R2 versus Adjusted R2


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R2 versus Adjusted R2

Such a measure is called Adj R2

If k > 1, Adj R2 R2, as the no of explanatory

variables increases in the model, Adj R2

becomes increasingly smaller than R2

It enable us to compare two models that havesame dependent variable but differentnumbers of independent variables

In our example, it can be shown that

Adj R2

=0.88 < 0.89 (R2

)

2 2 ( 1)1 (1 )( )

nR R

n k


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When to add an additional variable?

We often faced with problem ofdeciding among several competingexplanatory variables

Common practice is to add variables aslong as Adj R2 increases even though its

numerical value may be smaller than R2


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Computer output & Reporting


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The Chicken Consumption

Example

Explain US Consumption of Chicken

Time Series Observations - 1950-1984


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Variable Definitions

CHCONS - Chicken consumption in theUS

LDY - Log of disposable income in theUS

PC/PB - Price of Chicken relative to thePrice of Best Red Meat


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Data Time plots

Actual plots of the data over timefollows

Note the trends and cycles What are the relationships betweenthe variables?

Are movements in CHCONS related tomovements in LDY and PC/PB?

Time plot - CHCONS Actual Data


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0.0

10.0

20.0

30.0

40.0

50.0

60.0

1950

1952

1954

1956

1958

1960

1962

1964

1966

1968

1970

1972

1974

1976

1978

1980

1982

1984

CHCONS

YEAR

Timeplot-LDY Actual Data


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0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

7.0000

8.0000

9.0000

10.0000

L

DY

Year

Timeplot-PC/PB Actual Data


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0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

1.4000

1.6000

1950

1953

1956

1959

1962

1965

1968

1971

1974

1977

1980

1983

PC/PB

Year


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Chicken Consumption vs.

Income There may be a relationship betweenCHCONS and LDY

A simple plot of the two variablesseems to reveal this

Note the positive relationship

Scatter Plot - CHCONS vs. LYD


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0.0

10.0

20.0

30.0

40.0

50.0

60.0

7.0000 7.5000 8.0000 8.5000 9.0000 9.5000

CHCONS

LYD


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Chicken Consumption vs.

Relative Price of Chicken There may also be a relationshipbetween CHCONS and PC/PB

A plot of these two variables showsthe relationship

Note the negative relationship

Scatter Plot - CHCONS vs PC/PB


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0.0

10.0

20.0

30.0

40.0

50.0

60.0

0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 1.2000 1.4000 1.6000

CHCONS

PC/PB

CHCONS f(LDY)


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CHCONS = f(LDY)

Simple linear regression captures therelationship between CHCONS andLDY, assuming no other relationships

This regression explains much of thechange in CHCONS, but not everything

The plotted regression line shows thehypothesized relationship and theactual data


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CHCONS = f(LDY)LDY Const.

Coeff 15.86 -92.17SE(b) 0.53 4.34

R2 = 0.9641 SE(y) = 2.03F = 879.05 df = 33

SSReg= 3639.12 SSResid = 136.61(also called SSE) (also called SSR)

Regression Line - CHCONS = f(LYD)


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0.00

10.00

20.00

30.00

40.00

50.00

60.00

7.0000 7.5000 8.0000 8.5000 9.0000 9.5000

C

HCONS

LYD

CHCONS = f(LYD) Actual Data


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CHCONS = f(PC/PB)

Another simple regression examines therelationship between CHCONS and

PC/PB

While the line explains some of the

variation of CHCONS, there is moreunexplained error


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CHCONS = f(PC/PB)

PC/PB Const.Coeff -28.83 50.77SEb 2.93 1.75

R2 = 0.746 SE(y) = 5.39F = 97.14 df = 33SSReg = 2818.32 SSResid = 957.42

(also called ESS) (also called RSS)

Regression Line - CHCONS = f(PC/PB)


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0.00

10.00

20.00

30.00

40.00

50.00

60.00

0.0000 0.5000 1.0000 1.5000

CHCONS

PC/PB

CHCONS=f(PC/PB) Actual Data


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CHCONS = f(LDY,PC/PB)

LDY PC/PB Const.

Coeff 12.79 -8.08 -63.19

SEb 0.54 1.12 4.84

R2 = .986 SEy = 1.27

F = 1149.89 df = 32

SSReg = 3723.92 SSResid = 51.82

(SSE) (SSR)

Actual vs. Predicted


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0.0

10.0

20.0

30.0

40.0

50.0

60.0

CHCONS

YEAR

Actual CHCONS=f(LDY,PC/PB)

Table 7 8 Gujarati: US Defense budget outlays


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Table 7.8 Gujarati: US Defense budget outlays

1962 1981

Yt= Defense budget outlays for year t($ Bn)

X2t=GNP for year t($ Bn)

X3t=US military sales/assistance ($ Bn)X4t=Aerospace industry sales ($ Bn)

X5t= Military conflicts involving troops

=0, if troops < 100000=1, if troops > 100000

Table 8.10, Gujarati


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, j

Table gives data used by a telephone cablemanufacturer to predict sales to a majorconsumer for the period 1968 1983

Y=annual sales in MPF (million paired feet)

X2=GNP (billion $)X3=housing starts (1000 of units)X4=Unemployment rate (%)X5=Prime rate lagged 6 monthsX6= Customer line gains (%) Introduce later

Table 7 10 Gujarati


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Consider following demand function for money

in US for 1980 1998

Where, M = Real money demandY = Real GDP

r = Interest rate

LTRATE: Long term interest rate (30yr tr bond)

TBRATE: 3 months tr bill rate

tub

t

b

tt erYbM32

1

Regression Problems


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Regression Problems

Multicollinearity: two or more independentvariables are highly correlated, thus it isdifficult to separate the effect each has on thedependent variable.

Passing the F-test as a whole, but failing the t-test for each coefficient is a sign thatmulticollinearity exists.

A standard remedy is to drop one of theclosely related independent variables from theregression

Collinearity


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Collinearity

Y

X2

X3

Y

X2

X3

X3

X2 X3

Y



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Y= number of people employed (in 000)X1=GNP implicit price deflator

X2=Nominal GNP (million $)X3=number of people unemployed (in 000)X4=number of people in armed forceX5=Non-institutionalized population over 14 yearsX6=year=1for 1947, 2for 1948 and 16 for 1962

Regress and explain the results, Regress for shorter time-span, Pair wise correlation, Regress among Xs, Real GNP=(X2/X1),

Drop X6 as X5 & X6 are correlated Drop X3

Ex: Petrol demand=f[(car+two), teleden, price]

Regression Problem - Autocorrelation


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Regression Problem Autocorrelation

Lebanon Ex.

Errors are correlated

Observed in time series data

Say, output of a farm regress on capital and labor

on quarterly data and there is a labor strike on aparticular quarter affecting the output in that

particular quarter No autocorrelation

But if the strike affect the output in other quarters

as well Autocorrelation

Reasons


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Inertia or sluggishness

agricultural commoditiesSupplyt = B1+ B2Pt-1+ ut

Say at the end of period t, Pt turns out to be

lower than Pt-1, so the farmer may decode toproduce less in period (t+1)than t

Data manipulation Monthly to quarterly data by averaging,

thereby damping the fluctuating of monthlydata

Smoothness leads to systematic pattern indisturbances

Reasons


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Model Specification errors

Omitting relevant variable(s)

Ex: Y=b1+b2X2t+b3X3t+b4X4t+ut but we run theregressionY=b1+b2X2t+b3X3t+ vt, where

vt=b4X4t+ut If we plot v, it will not be random but exhibit a

systematic pattern creating (false)autocorrelation

Lagsconsumptiont=b1+b2incomet+b3consumptiont-1+ ut

A t l ti


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Autocorrelation

ut

Time

C f t l ti


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Consequences of autocorrelation

Least square estimates are linear andunbiased but they do not have minimumvariance property

t & F statistics are not reliable

R2 may be an unreliable measure of trueR2

Detection


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Detection

Plotting OLS residuals against time

No autocorrelation

errors are randomly distributed

Presence of autocorrelation errors exhibit a distinctbehavior

DW statistics (based on estimated residuals)

Others To correct autocorrelation consider:

Transforming the data into a different order ofmagnitude

Introducing lagged data

Ex: Carbon, petro-diesel


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Forecasting

Famous forecasting quotes


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g q

"I have seen the future and it is very much like the

present, only longer." - Kehlog Albran, The Profit This nugget of pseudo-philosophy is actually a concise

description of statistical forecasting. We search forstatisticalproperties of a time series that are constant in time - levels,trends, seasonal patterns, correlations and autocorrelations, etc.

We then predict that those properties will describe the future aswell as the present.

"Prediction is very difficult, especially if it's about thefuture."Niels Bohr, Nobel laureate in Physics

This quote serves as a warning of the importance of validating aforecasting model out-of-sample. It's often easy to find a modelthat fits the past data well--perhaps too well! - but quite anothermatter to find a model that correctly identifies those patterns inthe past data that will continue to hold in the future

Precise forecasting (demand prices etc) should


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Precise forecasting (demand, prices etc) should

be an integral part of the planning process

Billions in revenue can be lost if a company

forecast too low and its inventory is sold out

Similarly, a company can incur significant losses ifforecasts are too high and excess inventory builds

up

Thus, a comprehensive knowledge of theforecasting process is extremely important for

firms success and industrys sustenance


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There is an array of empirical methods that

are available today for forecasting

An appropriate method is chosen based on

the availability of the data and the desirednature of the forecasts

Long term (several years ahead)

Medium-term (quarterly to monthly) Short-term (daily to hourly to several minutes

ahead)

Forecasting Techniques


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Qualitative Analysis

Expert Opinion:

personal insight or panel consensus on future

expectations

Subjective in nature

Survey Methods

Through interview or questionnaires ask firms,

government agencies or individuals about their futureplans

Frequently supplement quantitative forecasts



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Econometric Forecasting

Combine economic theory and statistical tools to

predict future relations

Say, you have estimated following relationship

DDpetrol = 2.08 + 1.95 GDP 0.87 Prpetrol - 0.78 teldn

and you want to forecast petrol demand for

2016 using this relationship. SO

[DDpetrol ] 2016 = 2.08 + 1.95 GDP2016 0.87 [Prpetrol ] 2016- 0.78 [teldn]2016

We however need to know the future values of

independent variables



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Time Series Techniques

A time series is a sequence of observations takensequentially in time

An intrinsic feature of a time series is that,typically adjacent observations are dependent

The nature of this dependence amongobservations of a time series is of considerablepractical interest

Time Series Analysis is concerned with techniquesfor the analysis of this dependence

Time series data


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Time series data

Secular Trend: long run pattern

Cyclical Fluctuation: expansion and

contraction of overall economy (business

cycle)

Seasonality: annual sales patterns tied to

weather, traditions, customs

Irregular or random component

Trend & Cyclical Patterns


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0 2 4 6 8 10 12 14 16 18 20

Years(a)

Sales ($)

Secular trendCyclical patterns

Trend & Cyclical Patterns

Trend, Seasonal & Random


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Components

Long-run trend(secular plus cyclical)

peak

peak Seasonalpattern

Randomfluctuations

J F M A M J J A S O N DMonths

(b)

Sales ($)



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g q

Time Series Techniques

Examine the past behavior of a time series in

order to infer something about its future behavior

A sophisticated and widely used technique to

forecast the future demand

Examples

Univariate: AR, MA, ARMA, ARIMA, Exponential

Smoothing, ARIMA-GARCH etc.

Multivariate: VAR, Cointegration etc.

Ex-Post vs Ex-Ante Forecasts


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Ex-Post vs. Ex-Ante Forecasts

How can we compare the forecastperformance of our model?

There are two ways.

Ex Ante: Forecast into the future, wait for thefuture to arrive, and then compare the actual tothe predicted

Ex Post: Fit your model over a shortened sample Then forecast over a range of observed data

Then compare actual and predicted.

Ex-Post and Ex-Ante


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Estimation & Forecast Periods

Suppose you have data covering theperiod 1980.Q1-2001.Q4

Ex-PostEstimation Period

Ex-PostForecastPeriod

Ex-Ante

ForecastPeriod

TheFuture

Examining the In-Sample Fit


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g p

One thing that can be done, once youhave fit your model is to examine the in-sample fit

That is, over the period of estimation, you

can compare the actual to the fitted data

It can help to identify areas where yourmodel is consistently under or over

predicting

take appropriate measures Simply estimate equation and look at

residuals

Model Performance


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RMSE =(1/n(fi xi)2 - difference

between forecast and actual summed smaller the better

MAE & MAPE smaller the better

The Theil inequality coefficient alwayslies between zero and one, where zeroindicates a perfect fit.

Bias portion -Should be zero How far is the mean of the forecast from

the mean of the actual series?

Model Performance


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Variance portion - Should be zero How far is variation of forecast from forecast of

actual series variance?

Covariance portion - Should be one What portion of forecast error is unsystematic

(not predictable)

If your forecast is "good", the bias andvariance proportions should be small so thatmost of the bias should be concentrated on

Documents

3A Demand Estimationl