# Estimation Of Dynamic Consumption Function For Nigeria Economics Essay

This undertaking estimates the dynamic ingestion map for the state Nigeria from 1969 to 2003. In the survey, I focus on the additive dynamic specification in order to discourse the parametric quantity estimations in carry out assorted trials. The first chapter focal point on the ingestion map theory and how assorted economic system view assorted theory to demo the relationship between income and ingestion and how it can be specified in both additive and non additive signifier. The following chapter describe the information how it was obtained processed through Eviews and how it was interpreted to suit into the undertaking.

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The following chapter is hypothesis proving which is used to depict the significance of the guess and construction the relationship that determines dependent variables in ingestion. The t-test and F trial is used to mensurate the proportion in the dependent variables. it besides explain the important testing where we reject or Do non reject void hypothesis. It further estimates the long tally and short tally fringy leaning to devour and how they affect ingestion. it besides explain some trial such as Ramsey RESET trial, Heteroskadasticity trial, LM trial, for consecutive correlativity and trial for their misspecification.

Chapter 2: LITERATURE REVIEW

This chapter focal point on the manner the entire disposable income and entire ingestion and nest eggs are divided by a consumer.

The ingestion map is a mathematical expression laid out by the celebrated toilet Maynard Keynes. The expression was designed shown the relationship between existent disposal income and consumer disbursement. The map is used to cipher the sum of entire ingestion in an economic system. An independent ingestion is used to do up for ingestion that is non influenced by current income and induced ingestion that is influenced by the economic system ‘s degree of income.

The simple ingestion map is shown as the additive map:

Where:

C = entire ingestion,

c0 = independent ingestion ( c0 & gt ; 0 ) ,

c1 is the fringy leaning to devour ( ie the induced ingestion ) ( 0 & lt ; c1 & lt ; 1 ) , and

Yd = disposable income ( income after revenue enhancements and transportation payments, or W – Thymine ) .

Autonomous ingestion represents ingestion when income is zero. In appraisal, this is normally assumed to be positive. The fringy leaning to devour ( MPC ) , on the other manus measures the rate at which ingestion is altering when income is altering. In a geometric manner, the MPC is really the incline of the ingestion map.

The MPC is assumed to be positive. Therefore, as income additions, ingestion additions. However, Keynes mentioned that the additions ( for income and ingestion ) are non equal. Harmonizing to him, “ as income additions, ingestion additions but non by every bit much as the addition in income ” .

There are four attacks to consumption map and they are:

Absolute income hypothesis ( Keynes )

Relative income hypothesis ( James Duesenberry 1949 )

Permanent income hypothesis ( Friedman )

Life Cycle hypothesis ( Ando & A ; Modigliani )

A Absolute Income Hypothesis In economic science was proposed by English economic expert John Maynard Keynes ( 1883-1946 ) , and has been refined extensively during the sixtiess and 1970s, notably by American economic expert James Tobin ( 1918-2002 ) .

The theory examines the relationship between income and ingestion, and asserts that the ingestion degree of a family depends non on its comparative income but on its absolute degree of income. As income rises, the theory asserts, ingestion will besides lift but non needfully at the same rate.

Developed by James Stemble Duesenberry, the comparative income hypothesis provinces that an person ‘s attitude to ingestion and economy is dictated more by his income in relation to others than by abstract criterion of life. So an person is less concerned with absolute degree of ingestion than by comparative degrees. The per centum of income consumed by an single depends on his percentile place within the income distribution.

Second it hypothesizes that that present ingestion is non influenced simply by present degrees of absolute and comparative income, but besides by degrees of ingestion attained in old period. It is hard for a household to cut down a degree of ingestion one time attained. The aggregative ratio of ingestion to income is assumed to depend on the degree of present income relation to past peak income.

The current degree of ingestion is a straightforward map, driven by the current degree of income. This implies that people adapt outright to income alterations.

– There is rapid version to income alterations

– The snap of ingestion to current income alterations

The snap with regard to current income in other theories will be less. They cut down the sensitiveness to current income flows.

Permanent income hypothesis ( PIH ) This theory of ingestion that was developed by the American economic expert Milton Friedman. In its simplest signifier, the hypothesis states that the picks made by consumers sing their ingestion forms are determined non by current income but by their longer-term income outlooks. The cardinal decision of this theory is that transitory, short-run alterations in income have small consequence on consumer disbursement behaviour.

Measured income and measured ingestion contain a lasting ( anticipated and planned ) component and a transitory ( windfall gain/unexpected ) component. Friedman concluded that the person will devour a changeless proportion of his/her lasting income ; and that low income earners have a higher leaning to devour ; and high income earners have a higher ephemeral component to their income and a lower than mean leaning to devour.

In Friedman ‘s lasting income hypothesis theoretical account, the cardinal determiner of ingestion is an person ‘s existent wealth, non his current existent disposable income. Permanent income is determined by a consumer ‘s assets ; both physical ( portions, bonds, belongings ) and human ( instruction and experience ) . These influence the consumer ‘s ability to gain income. The consumer can so do an appraisal of awaited lifetime income. He besides explains why there was no bead prostration in disbursement station WWI. Friedman argues that it would be more reasonable for people to utilize current income, but besides at the same clip to organize outlooks about future degrees of income and the comparative sums of hazard.

Therefore, they are organizing an analysis of “ lasting income. ”

Permanent Income = Past Income + Expected Future Income

Ephemeral Income – income that is earned in surplus of, or perceived as an unexpected windfall. If you get income non equal to what you expected or to what you do n’t anticipate to acquire once more.

So, he argues that we tend to pass more out of lasting income than out of transitory.

In the Friedman analysis, he treats people as organizing their degree of expected future income based on their past incomes. This is known as adaptative outlooks.

Adaptive Expectations – looking frontward in clip utilizing past outlooks. In this instance, we use a distributed slowdown of past income.

YPt+1 A® E ( Yt+1 ) = B0Yt + B1Yt-1 + B2Yt-2aˆ¦

Where B0 & gt ; B1 & gt ; B2

It is besides possible to add a restraint: B0 + B1 + B2 + B3 + aˆ¦ Bn = 1 this is expected income, the existent income can be thought of as: Yt-1 – Ypt+1 = Ytt+1 Using this, we can build a new theoretical account of the ingestion map: A Ct = a = bYDt + cYtt

There are other factors that people can look at to believe about future degrees of income. For illustration, people can believe about future involvement rates and their consequence on their income watercourse.

The Relative Income Hypothesis The Duesenberry attack says that people are non merely concerned about absolute degrees of ownership. They are in fact concerned about their ownerships relative to others. Peoples are non needfully happier if they have more money. They do nevertheless describe higher felicity if they have more comparative to others. The new public-service corporation map would be:

Current economic experts still back up this thought. Ex-husband: Robert Frank and Juliet Schor

Duesenberry argues that we have a greater inclination to defy disbursement lessenings relative to falls in income than we do to increase outgo comparative to additions of income. The ground is that we do n’t desire to change our criterion of populating downard.

CT = a +bYT + cYX

YX is the old peak degree of income ( this keeps outgo from falling in the face of income beads ) . It is besides known as the Drag Effect.

A displacement in outgos relative to a old degree of income is known as the Ratchet Effect, and will be shown below.

Duesenberry argues that we will switch the curve up or travel along the curve, but non we will defy displacements down. When WWII ended, a important figure of economic experts claimed that there would be a ingestion diminution and aggregative demand bead which did non happen. This provides back uping grounds.

A long-term ingestion map can be drawn, presuming that there is a growing tendency. If this is true, old peak income would hold been that of last twelvemonth and therefore would give a ingestion map that looks like it depends on current income.

## A The Life Cycle Hypothesis

This is chiefly attributed to Ando and Modigliani

The basic impression is that ingestion disbursement will be smooth in the face of an fickle watercourse of income

Working Phase:

Maintain current ingestion, pay off debt from youth old ages

Maintain current ingestion, construct up militias

Age distribution now matters when we look at ingestion, and in general, the leaning to devour. Debt and wealth are besides taken into history when we look at the leaning to devour. The dependance construction of the population will impact or act upon ingestion forms.

Lester Thurow ( 1976 ) – argued that this theoretical account does n’t work because it does n’t assume there is any motivation for edifice wealth other than ingestion. Thurow argues that their existent motive is position and power ( both internal and external to the household ) . The lasting income hypothesis bears a resemblance to the life-cycle hypothesis in that in some sense, in both hypotheses, the persons must act as if they have some sense of the hereafter.

Chapter 3: Data

This chapter explain the informations used in this undertaking. The state used here is Nigeria and the information is downloaded from the IMF International Financial Statistics. The state tabular array is chosen from Annual IFS series via beyond 20/20 WDS. From the Nigeria Annual IFS series the undermentioned variables were selected: 96F.CZF HOUSEH.CONS.EXPEND. INCL.NPISHS SA ( Unit of measurements: National Currency )

( Scale: Billions ) = NCONS

64… ZF CPI: ALL ITEMS ( 2000=100 ) ( Unit of measurements: Index Number ) = CPI

99I.CZF GROSS NATIONAL DISPOSABLE INCOME SA ( Unit of measurements: National Currency )

( Scale: Billions ) = NYD

99BIRZF GDP DEFLATOR ( 2000=100 ) ( Unit of measurements: Index Number ) = GDPdef N ) = NCONS

64… ZF CPI: ALL ITEMS ( 2000=100 ) ( Unit of measurements: Index Number ) = CPI

99I.CZF GROSS NATIONAL DISPOSABLE INCOME SA ( Unit of measurements: National Currency )

( Scale: Billions ) = NYD

99BIRZF GDP DEFLATOR ( 2000=100 ) ( Unit of measurements: Index Number ) = GDPdef

Then chink on show tabular arraies, from Download Icon, choice Microsoft excel format.

From Excel, I did change over nominal series to existent series by utilizing a method called deflating by monetary value Index.

Rcons, which is the Real Consumption outgo series for USA, is obtain by spliting the NCONS ( 96F.CZF… ) : Family ingestion outgo by CPI ( 64… ZF )

## Rcons

RYD is the Real Disposable Income for USA, which is obtain by deflating the Nominal Disposable Income ( NYD ) 99I.CZF by the GDP deflator ( 99BIRZF ) as,

## RYD=

Then the consequences are saved in a new spreadsheet and are so sent in EViews.

Chapter 4: ESTIMATED RESULT AND INTERPRETATION

This chapter reports the ordinary least squares result used for Nigeria between 1969 and 2003 specified both additive and log additive signifiers.

Interpretation of the intercept is explained as when the information series lies far from the beginning of the X, Y secret plan of the variable, estimations of the will non hold a meaningful economic system theory reading.

Fringy leaning and Elasticities.

— — — — — — – Linear specification

Estimated coefficient are fringy leanings, the proportion of a ( N ) income that will pass on ingestion.

Where N is the unit of measuring

— — — — — — Log additive specification.

The estimated coefficient from a additive specification are non fringy leanings, they are snaps because they measure the reactivity of the independent variable alterations in the independent variables. Therefore, snaps of 0.70 will bespeak that 70 % of alteration in independent variable will increase the dependent by 70 % .

Elasticity is really good because it tends to associate to run while the leanings tends to associate to the norm.

## Coverage of dynamic additive specification:

Dynamic additive specification: rcons c ryd rcons ( -1 ) ) :

{ 4.506 } { 0.20 } { 0.74 } [ Coefficient ]

{ 5.375 } { 0.14 } { 0.18 } [ Std. Error ]

{ 0.838 } { 1.38 } { 4.10 } [ t-Statistic ]

{ 0.408 } { 0.18 } { 0.00 } [ Prob ]

## Coverage of dynamic log additive specification:

Dynamic log additive specification: log ( rcons ) c log ( ryd ) log ( rcons ( -1 ) )

Report of dynamic log additive specification

{ 0.951 } { 0.163 } { 0.598 } [ Coefficient ]

{ 0.380 } { 0.097 } { 0.164 } [ Std. Error ]

{ 2.501 } { 1.677 } { 3.638 } [ t-Statistic ]

{ 0.018 } { 0.104 } { 0.001 } [ Prob ]

Testing that the MPC of coefficients of Ryd and and Zero.

4.506 + 0.20 Ryd + 0.74

{ 5.375 } { 0.14 } { 0.18 }

H0: I?= 0

H1: I? a‰ 0

tstatistics= ==1.43

tcrit = tn-k, I±/2

Where,

n=34

k=3,

=5 %

t=34-3, 0.05/2 = T 31, 0.025

## 0.025

We will hence non reject the void hypothesis that MPC coefficient of Ryd ( ) is equal to zero because 1.432.042 which besides means that ordinary least squares of is non statistically important to zero.

4.506 + 0.20 Ryd + 0.74

{ 5.375 } { 0.14 } { 0.18 }

H0: = 0

H1: a‰ 0

T-statistics= ==4.11

tcrit = tn-k, I±/2

Where n=34, k=3, =5 %

t=34-3, 0.05/2=t 31, 0.025

We will therefore reject the void hypothesis that MPC of the coefficient of Ryd ( ) is equal to zero because 4.112.042 which besides means that ordinary least squares of is statistically important to zero.

Testing that the Elasticities of coefficients of Ryd and and Zero.

H0: I?= 0

H1: I? a‰ 0

t-statistics= ==1.43

tcrit = tn-k, I±/2

where n=34, k=3, =5 %

t=34-3, 0.05/2=t 30,0.025

## 0.025

We will hence non reject the void hypothesis that MPC coefficient of Ryd ( ) is equal to zero because 1.432.042 which besides means that ordinary least squares of is non statistically important to zero.

4.506 + 0.20 Ryd + 0.74

{ 5.375 } { 0.14 } { 0.18 }

H0: = 0

H1: a‰ 0

T-statistics= ==4.11

tcrit = tn-k, I±/2

Where n=34, k=3, =5 %

t=34-3, 0.05/2=t 30, 0.025

Chapter 5: Testing THE SIGNIFICANCE OF THE ESTIMATED

Parameter.

A statistical hypothesis trial is a method of doing statistical determinations utilizing experimental informations. In statistics, a consequence is called statistically important if it is improbable to hold occurred by opportunity. The phrase “ trial of significance ” was coined by Ronald Fisher: “ Critical trials of this sort may be called trials of significance, and when such trials are available we may detect whether a 2nd sample is or is non significantly different from the first.

Hypothesis testing is sometimes called confirmatory informations analysis, in contrast to exploratory informations analysis. In frequence chance, these determinations are about ever made utilizing null-hypothesis trials ; that is, 1s that answer the inquiry Assuming that the void hypothesis is true, what is the chance of detecting a value for the trial statistic that is at least every bit utmost as the value that was really observed? One usage of hypothesis testing is make up one’s minding whether experimental consequences contain adequate information to project uncertainty on conventional wisdom.

Statistical hypothesis testing is a cardinal technique of frequents statistical illation, and is widely used, but besides much criticized. The chief direct option to statistical hypothesis testing is Bayesian illation. However, other attacks to making a determination based on informations are available via determination theory and optimum determinations.

The critical part of a hypothesis trial is the set of all results which, if they occur, will take us to make up one’s mind that there is a difference. That is, do the void hypothesis to be rejected in favour of the alternate hypothesis. The critical part is normally denoted by C.

Null Hypothesis is a phrase that was originally coined by English geneticist and statistician Ronald Fisher. In statistical hypothesis testing, the void hypothesis ( H0 ) officially describes some facet of the statistical “ behavior ” of a set of informations. This description is assumed to be valid unless the existent behavior of the information contradicts this premise. Therefore, the void hypothesis is contrasted against another or alternate hypothesis. Statistical hypothesis testing, which involves a figure of stairss, is used to make up one’s mind whether the informations contradicts the void hypothesis. This is called significance testing. A void hypothesis is ne’er proven by such methods, as the absence of grounds against the void hypothesis does non set up its truth. In other words, one may either cull, or non reject the void hypothesis ; one can non accept it. This means that one can non do determinations or pull decisions that assume the truth of the void hypothesis. Just as neglecting to reject it does non “ turn out ” the void hypothesis, one does non reason that the alternate hypothesis is disproven or rejected, even though this seems sensible. One merely concludes that the void hypothesis is non rejected. Not rejecting the void hypothesis still allows for acquiring new informations to prove the alternate hypothesis once more. On the other manus, rejecting the void hypothesis merely means that the alternate hypothesis may be true, pending farther testing.

## Test the overall significance of Dynamic additive arrested development theoretical account ingestion map.

4.506 + 0.20 Ryd + 0.74

{ 5.375 } { 0.14 } { 0.18 }

R2=0.70

N=34

Holmium: I? = I? = 0

H1: I? a‰ I? a‰ 0

0.05

0.95

We will therefore reject the void hypothesis that MPC coefficients of Ryd ( ) and ( ) is equal to zero because Fs & gt ; Fc ( 36.16 & gt ; 3.32 ) which besides means that ordinary least squares of is statistically important, that is, it is non zero.

Intercept is non statistically important at the 5 % degree in any instance ; in appraisal the intercept has no economic significance.

Interpretation — – graduated table — – factor.

The coefficient on RYD and the lagged dependant variable ( LDV ) statistically important at 5 %

The Regression on coefficient of the finding R2

R2 is a step of the proportion of the fluctuation in the dependant variable that is explained by the full variable in the equation.

In statistics, the coefficient of finding, R2 is used in the context of statistical theoretical accounts whose chief intent is the anticipation of future results on the footing of other related information. It is the proportion of variableness in a information set that is accounted for by the statistical theoretical account. It provides a step of how good future results are likely to be predicted by the theoretical account.

There are several different definitions of R2 which are merely sometimes tantamount. One category of such instances includes that of additive arrested development. In this instance, R2 is merely the square of the sample correlativity coefficient between the results and their predicted values, or in the instance of simple additive arrested development, between the result and the values being used for anticipation. In such instances, the values vary from 0 to 1. Important instances where the computational definition of R2 can give negative values, depending on the definition used, originate where the anticipations which are being compared to the corresponding result have non derived from a model-fitting process utilizing those informations.

R2 is expressed as follows:

R2 = ESS/TSS = 1 – RSS/TSS = 1 -a?‘e2i /a?‘ ( Yi – Y- ) 2

Where ; ESS= Explained amount of squares

TSS= Total amount of squares

Therefore R2 in dynamic additive arrested development theoretical account is explained as follows.

R2 is the coefficient of finding. It is besides used to mensurate to mensurate the goodness of tantrum. It is besides used to state us how close the information point are to the fitted map that is 0 & lt ; R2 & lt ; 1. More exactly 0.70 in dynamic additive arrested development theoretical account ingestion map tell us the 70 % of the variableness in Rconst can be explained by the variableness in the explanatory variables ( Ryd and Rconst-1 ) .

Appraisal OF LONG RUN AND SHORT RUN MARGINAL PROPENSITY TO CONSUME

The two different period of MPC. The first 1 is the Short Run Marginal consume and it shows us the fringy leaning to devour. This shows the fringy leaning to devour for one period of clip.

This simple indicates that a 1 unit alteration in disposable income would hold on ingestion in the same period.

The 2nd shows in the Long Run Marginal Propensity to Consume, it takes into consideration recent ingestion behavior, every bit good as disposable income when finding the degree of ingestion.

By taking into consideration the old ingestion, and the current income, it allows you to measure what consequence the old could likely hold on ingestion.

Long tally =

Long Run is obtained through the usage of steady province ingestion. We can presume that

it allows dynamic equation to be written as

Taking to Left Hand Side we get

When C is factored out it is written as:

When work outing for C we obtain

C =

So utilizing the SSQ we get the Long Run where C is the ingestion and is being determine by the Long run Marginal Propensity to Consume of the income variable.

Chapter 6: THE RAMSEY RESET Trial

Ramsey Regression Equation Specification Error Test ( RESET ) trial ( Ramsey, 1969 ) is a general specification trial for the additive arrested development theoretical account. More specifically, it tests whether non-linear combinations of the estimated values help explicate the endogenous variable. The intuition behind the trial is that, if non-linear combinations of the explanatory variables have any power in explicating the endogenous variable, so the theoretical account is mis-specified. The RESET trial is designed to observe omitted variables and wrong functional signifier.

The Ramsey trial so tests whether ( I?1x ) 2, ( I?2x ) 3… , ( I?k a?’ 1x ) K has any power in explicating Y. This is executed by gauging the followers and so proving, by a agency of a F-test whether through are zero. If the null-hypothesis that all arrested development coefficients of the non-linear footings are zero is rejected, so the theoretical account suffers from mis-specification.

For a univariate ten the trial can besides be performed by regressing on the abbreviated power series of the explanatory variable and utilizing an F-Test for

Test rejection implies the same penetration as the first version mentioned above.

The F-test compares arrested developments, the original one and the Ramsey ‘s subsidiary one, as done with the rating of additive limitations. The original theoretical account is the restricted one opposed to the Ramsey ‘s unrestricted theoretical account.

F ( k a?’ 1, n a?’ K ) , where:

N is the sample size ;

K is figure of parametric quantities in the in the Ramsey ‘s theoretical account.

Furthermore, the additive theoretical account and the theoretical account with the non-linear power footings are subjected to the F-test, likewise as before:

~ F ( k a?’ 1, n a?’ m a?’ K ) ,

where m + K is figure of parametric quantities in the Ramsey ‘s theoretical account, which are thousand a?’ 1 variables in the Ramsey group ( non-linear )

plus thousand + 1 the figure of parametric quantities in the original theoretical account

Trial for misspecification.

Rejection of Ho implies the original theoretical account is unequal and can be improved. A failure to reject H0 says the trial has non been able to observe any misspecification.

Overall, the general doctrine of the trial is: If we can significantly better the theoretical account by unnaturally including powers of the anticipations of the theoretical account, so the original theoretical account must hold been unequal.

## iˆ

Estimating this theoretical account, and so augmenting it with squares of the anticipations, and squares and regular hexahedrons of so anticipations, outputs the RESET trial consequences The F-values are rather little and their corresponding p-values of 0.93 and 0.70 are good above the conventional significance degree of 0.05. There is no grounds from the RESET trial to propose the log-log theoretical account is unequal.

Chapter 7: BREUSCH- GODFERYSERIAL CORRELATION LM Trial

The Breusch – Godfery Lagrange Multiplier ( BGLM ) Test. computes the Breusch ( 1978 ) -Godfrey ( 1978 ) Lagrange multiplier trial for no independency in the mistake distribution. For a specified figure of lags P, the trial ‘s nothing of independent mistakes has options. The trial statistic, a T R^2 step, is distributed Chi-squared ( P ) under the void hypothesis.

In statistics, the BG consecutive correlativity LM trial is a large trial for autocorrelation in the remainders from a arrested development analysis and is considered more general than the standard Durbin-Watson statistic. The void hypothesis is that there is no consecutive correlativity of any order up to p.un like the Durbin-Watson H statistic is merely valid for nonstochastic regressors and first-order autoregressive strategies, the BG trial has none of these limitations, and is statistically more powerful than Durbin ‘s H statistic.

Features of Breusch-Godfrey Test:

Allows for a relationship between Greenwich Mean Time and several of its slowdowns

Estimate a arrested development and obtain remainders

Regress remainders on all regressors and lagged remainders

Obtain R2 from this arrested development

Leting T denote the figure of observations.

The job of clip series is frequently a job when the stochastic term in one period is non the independent in another.

Consecutive correlativity does non impact the unbiasedness of the belongingss of ordinary least squared but it does impact the minimal discrepancy hence, the estimated efficiency.

The LM trial uses the signifier that in order to see if the estimated remainder from the restricted signifiers is related to the lagged values of itself so

If consecutive correlativity does non be, the so the reasoning backwards of the unrestricted signifier and restricted signifier are the same.

There two signifiers of the trial reported and they are:

( 1 ) Chi-squared ( i??2 ) : the trial statistics is calculated as TR2, where T is the figure of observations in the original arrested development and R2 is the R-squared in the subsidiary arrested development. This has ai??2 ( H ) distribution for H limitations ( lags ) . So where h=5.

If TR2 & lt ; ( i??2 ) : ( H ) 0.05 so we do non reject the nothing of no autocorrelation ( at the 5 % Significance degree ) .

If TR2 & gt ; i??2 ( H ) 0.05 so we must reject the nothing of no autocorrelation

( At the 5 % Significance degree ) .

( 2 ) F Trial

The trial statistic is calculated as Fcal = ( T-k-1-h ) R2/h ( 1-R2 ) , where K is the figure of regressors in the original equation ( here k = 3 ) . This has an F ( H, T-k-1-h ) distribution.

If Fcal & lt ; F tabular arraies, 0.05 so we do non reject the nothing of no autocorrelation ( at the 5 %

Significance degree ) .

If Fcal & gt ; F tabular arraies, 0.05 so we must reject the void hypothesis of no autocorrelation ( at the 5 % significance degree )

The Chi- squared is traveling to be used in transporting out the trial below

## .

HETEROSKEDASTICITY

One of the premises of the classical additive arrested developments ( CLR ) theoretical account is that the error term in the additive theoretical account is drawn from a distribution with a changeless discrepancy when this is the instance, it is said that the mistakes are homoskedastic. In a state of affairs where this premise does non keep, so the job of heteroskedasticity occurs.

Heteroskedasticity, as a misdemeanor of the premises of the CLR theoretical account, causes the OLS estimations to lose some of their nice theoretical account.

Heteroskadasticity is likely to take topographic point in a cross- sectional theoretical account than clip series theoretical account.

Heteroskedasticity causes OLS to undervalue the discrepancies and standard mistakes of the estimated coefficients

This implies that the t-test and F-test are non dependable

The t-statistics tend to be higher taking us to reject a void hypothesis that should non be rejected

F-statistic follows an F distribution with K grades of freedom in the numerator and ( n – K -1 ) grades of freedom in the denominator

Reject the void hypothesis that there exists no heteroskedasticity if the F-statistic is greater than the critical F-value at the selected degree of significance.

If the nothing can non be rejected, so there exists heteroskedasticity in the information and an alternate appraisal method to OLS must be followed.

When proving for heteroskadasticity, there different ways to travel because heteroskadasticy takes a figure of different signifiers and it exact manifestation in a given equation is about known.

The chief focal point will be on the white trial which is more by and large used than the other trials.

The white trial is used to observe the heteroskadasticity by running a arrested development with the squared remainders as the dependant variable. In the right manus side of the 2nd equation all the original independent variables, the squared of all the original independent variables with each other and the transverse merchandise of the full original trial with each other. The white trial run the advantage of non presuming any peculiar signifier of heteroskadasticity which make it popular as one of the best trials yet to devised to use to all types of heteroskadasticity.

F-statistic follows an F distribution with K grades of freedom in the numerator and ( n – K -1 ) grades of freedom in the denominator

Reject the void hypothesis that there exists no heteroskedasticity if the F-statistic is greater than the critical F-value at the selected degree of significance.

If the nothing can non be rejected, so there exists heteroskedasticity in the information and an alternate appraisal method to OLS must be followed.

When proving for heteroskadasticity, there different ways to travel because heteroskadasticy takes a figure of different signifiers and it exact manifestation in a given equation is about known.

The chief focal point will be on the white trial which is more by and large used than the other trials.

The white trial is used to observe the heteroskadasticity by running a arrested development with the squared remainders as the dependant variable. In the right manus side of the 2nd equation all the original independent variables, the squared of all the original independent variables with each other and the transverse merchandise of the full original trial with each other. The white trial run the advantage of non presuming any peculiar signifier of heteroskadasticity which make it popular as one of the best trials yet to devised to use to all types of heteroskadasticity.

This is a White trial for heteroskedasticity with cross-products.

The trial statistics is a with process ( where H = k-1 from the subsidiary equation ( in this instance h=6-1=5 ) . We obtain critical values for the trial by utilizing the tabular arraies. For case making the trial at the 5 % important degree with h=5, five grades of freedom, would connote a trial statistic of.

With the trial critical value available we would so cipher the trial statistic and this is from the subsidiary equation. So from the subsidiary equation we had T=34 observations and from the one we obtained an R2 = 0.374449, the trial statistics would be

The trial statistics NR2 = TR2 = 34 ( 0.374449 ) = 12.731266

Since the we will so reject the void hypothesis homoskedasticity

Chapter 10: STATIONERITY, RANDOM WALK AND COINTEGRATION

A common premise in many clip series techniques is that the informations are stationary.

A stationary procedure has the belongings that the mean, discrepancy and autocorrelation construction do non alter over clip. Stationarity can be defined in precise mathematical footings, but for our intent we mean a level looking series, without tendency, changeless discrepancy over clip, a changeless autocorrelation construction over clip and no periodic fluctuations.

Non-stationary informations are unpredictable and can non be modelled or forecasted. The consequences obtained by utilizing non-stationary clip series may be specious in that theyA may bespeak a relationship between two variables where oneA does non be. In order to have consistent, dependable consequences, the non-stationary informations demands to be transformed into stationary informations. In contrast to the non-stationary procedure that has a variable discrepancy andA a mean that does non stay close, or returns to a long-term mean over clip, the stationary procedure reverts around a changeless long-run mean and has a changeless discrepancy independent of clip.

## Transformations to Achieve Stationarity

If the clip series is non stationary, we can frequently transform it to stationarity with one of the following techniques.

We can difference the information. That is, given the series Zt, we create the new series

The differenced informations will incorporate one less point than the original informations. Although you can difference the informations more than one time, one difference is normally sufficient.

If the information contain a tendency, we can suit some type of curve to the informations and so pattern the remainders from that tantrum. Since the intent of the tantrum is to merely take long term tendency, a simple tantrum, such as a consecutive line, is typically used.

For non-constant discrepancy, taking the logarithm or square root of the series may stabilise the discrepancy. For negative informations, you can add a suited invariable to do all the informations positive before using the transmutation. This changeless can so be subtracted from the theoretical account to obtain predicted that is, the fitted values and prognosiss for future points.

RANDOM WALK

A random walk is defined as a procedure where the current value of a variable is composed of the past value plus an error term defined as a white noise ( a normal variable with nothing mean and discrepancy one ) .

Algebraically a random walk is represented as follows:

The deduction of a procedure of this type is that the best anticipation of Y for following period is the current value, or in other words the procedure does non let foretelling the alteration. That is, the alteration of Y is wholly random.

It can be shown that the mean of a random walk procedure is changeless but its discrepancy is non. Therefore a random walk procedure is non stationary, and its discrepancy additions with T.

In pattern, the presence of a random walk procedure makes the prognosis procedure really simple since all the future values of, for s & gt ; 0, is merely.

## A random walk theoretical account with impetus

A impetus acts like a tendency, and the procedure has the undermentioned signifier:

For & gt ; 0 the procedure will demo an upward tendency.

Assuming a = 1. This procedure shows both a deterministic tendency and a stochastic tendency. Using the general solution for the old procedure,

Where is the deterministic tendency and is the stochastic tendency.

The relevancy of the random walk theoretical account is that many economic clip series follow a form that resembles a tendency theoretical account. Furthermore, if two clip series are independent random walk processes so the relationship between the two does non hold an economic significance. If one still estimates a arrested development theoretical account between the two the undermentioned consequences are expected:

( a ) High R2

( B ) Low Durbin-Watson statistic ( or high )

( degree Celsius ) High T ratio for the incline coefficient

This indicates that the consequences from the arrested development are specious. A arrested development in footings of the alterations can supply grounds against old specious consequences. If the coefficient of a arrested development

## a?†a?†

is non important, so this is an indicant that the relationship between Y and is specious, and one should continue by choosing other explanatory variable.

If the Durbin-Watson trial is passed so the two series are co- incorporate, and a arrested development between them is appropriate.

## COINTEGRATION

Cointegration theory is decidedly the invention in theoretical econometrics that has created the most involvement among economists.. The definition in the simple instance of 2 clip series xt and yt, that are both integrated of order one ( this is abbreviated I ( 1 ) , and means that the procedure contains a unit root ) , is the followers:

Definition

crosstalk and yt are said to be cointegrated if there is being of a parametric quantity I± such that

is a stationary procedure.

This turns out to be a way interrupting manner of looking at clip series, because it seems that tonss of economic series behaves that manner and because this is frequently predicted by theory.

The first thing to notice is that economic series behave like I ( 1 ) procedures, i.e. they

seem to float all over the topographic point, but the 2nd thing to notice is that they seem to float in

such a manner that the they do non float off from each other. If this is formulated statistically

it comes up with the cointegration theoretical account.

## Testing FOR RANDOM WALK

A unit root trial is normally carried out by utilizing the arrested development trial introduced by Dickey and Fuller ( 1979 ) . Under the void hypothesis the series should be a random walk. But a non-stationary series can normally be decomposed into a random walk and a stationary constituent.

## The Dickey-Fuller Unit Root Test for Non-stationarity

A Dickey-Fuller trial is an econometric trial for whether a certain sort of clip series informations has an autoregressive unit root. In peculiar in the clip series econometric theoretical account Y [ T ] = by [ t-1 ] + vitamin E [ T ] , where T is an whole number greater than zero indexing clip, and b=1, allow bOLS denote the OLS estimation of B from a peculiar sample. Let T be the sample size. Then the trial statistic T* ( bOLS -1 ) has a known, documented distribution. Its value in a peculiar sample can be compared to that distribution to find a chance that the original sample came from a unit root autoregressive procedure ; that is, one in which b=1.

## Augmented Dickey Fuller trial

An augmented Dickey-Fuller trial is a trial for a unit root in a clip series sample. An augmented Dickey-Fuller trial is a version of the Dickey-Fuller trial for a larger and more complicated set of clip series theoretical accounts. The augmented Dickey-Fuller ( ADF ) statistic, used in the trial, is a negative figure. The more negative it is, the stronger the rejections of the hypothesis that there is a unit root at some degree of assurance.

Appendixs

APPENDIX 1

## 99BIPZF GDP DEFLATOR ( 2003=100 ) ( Unit of measurements: Index Number )

1969

0.142026

2.901

3.682

0.381415

1970

0.161565

4.143

5.125

0.576998

1971

0.187414

5.09

6.853

0.584776

1972

0.193894

5.267

7.133

0.601693

1973

0.204369

6.903

10.578

0.324827

1974

0.230272

10.962

18.376

0.463547

1975

0.308482

13.689

21.559

0.559422

1976

0.383443

16.297

27.298

0.629019

1977

0.441296

19.061

32.272

0.682775

1978

0.537098

24.341

35.61

0.790858

1979

0.599991

25.928

42.535

0.918183

1980

0.659824

31.695

49.759

1.0211

1981

0.797151

34.563

49.839

1.12529

1982

0.858514

36.284

50.547

1.15585

1983

1.0578

41.457

56.168

1.3435

1984

1.2463

47.962

62.009

1.57577

1985

1.33897

54.066

70.732

1.63874

1986

1.41552

56.204

68.682

1.60447

1987

1.57533

78.329

97.225

2.40248

1988

2.43407

113.073

132.503

2.91572

1989

3.66246

138.828

207.173

4.20235

1990

3.93218

155.274

238.27

4.49293

1991

4.44364

222.27

299.536

5.33186

1992

6.425

404.182

485.404

8.79247

1993

10.0979

537.473

627.911

10.9303

1994

15.8569

694.053

849.231

14.0805

1995

27.4063

1543.09

1773.65

29.7889

1996

35.4276

2367.96

2612.84

40.9551

1997

38.4496

2434.62

2713.96

41.2998

1998

42.2931

2757

2705.42

39.5653

1999

45.0923

1969.37

3082.41

44.3461

2000

48.2186

2446.54

4618.55

64.1422

2001

57.3193

3642.58

4517.46

60.105

2002

64.7

5540.19

5216.64

84.6155

2003

73.7786

7044.54

6763.74

100

APPENDIX 2

Time

RCONS

RYD

1969

20.42584

9.653527

1970

25.64293

8.88218

1971

27.15912

11.71902

1972

27.16433

11.85488

1973

33.77714

32.56503

1974

47.60457

39.64215

1975

44.37536

38.53799

1976

42.50175

43.39774

1977

43.19323

47.26594

1978

45.31948

45.02705

1979

43.21398

46.32519

1980

48.03554

48.73078

1981

43.35816

44.28992

1982

42.26373

43.73145

1983

39.19172

41.80722

1984

38.48351

39.35156

1985

40.3788

43.16243

1986

39.70555

42.80666

1987

49.72228

40.4686

1988

46.45429

45.44435

1989

37.90567

49.29932

1990

39.48802

53.03221

1991

50.0198

56.17852

1992

62.9077

55.20678

1993

53.22622

57.44682

1994

43.76978

60.31256

1995

56.30421

59.54063

1996

66.83941

63.79767

1997

63.31977

65.71364

1998

65.18794

68.3786

1999

43.6742

69.50803

2000

50.73851

72.00486

2001

63.54893

75.15947

2002

85.6289

61.65112

2003

95.48216

67.6374

APPENDIX 3

LINEAR REGRESSION MODEL CONSUMPTION FUNCTION

Linear specification: rcons c ryd

Linear equation: rcons = I± + I?ryd

Dependent Variable: RCONS

Method: Least Squares

Date: 10/26/09 Time: 14:23

Sample: 1969 2003

Included observations: 35

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

15.33550

5.088069

3.014011

0.0049

RYD

0.680475

0.100993

6.737814

0.0000

R-squared

0.579072

A A A A Mean dependant volt-ampere

47.60036

0.566316

A A A A S.D. dependant volt-ampere

15.44940

S.E. of arrested development

10.17415

A A A A Akaike info standard

7.533023

Sum squared resid

3415.940

A A A A Schwarz standard

7.621900

Log likeliness

-129.8279

A A A A Hannan-Quinn criter.

7.563703

F-statistic

45.39813

A A A A Durbin-Watson stat

0.867793

Prob ( F-statistic )

0.000000

DYNAMIC LINEAR SPECIFICATION

LOG Linear specification: log rcons c logryd

LOG LINEAR EQUATION: I± + log I?ryd

Dependent Variable: LOG ( RCONS )

Method: Least Squares

Date: 10/21/09 Time: 15:04

Sample: 1969 2003

Included observations: 35

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

2.039330

0.211973

9.620723

0.0000

LOG ( RYD )

0.473350

0.055936

8.462289

0.0000

R-squared

0.684544

A A A A Mean dependant volt-ampere

3.814441

0.674984

A A A A S.D. dependant volt-ampere

0.316484

S.E. of arrested development

0.180428

A A A A Akaike info standard

-0.531526

Sum squared resid

1.074289

A A A A Schwarz standard

-0.442649

Log likeliness

11.30171

A A A A Hannan-Quinn criter.

-0.500846

F-statistic

71.61033

A A A A Durbin-Watson stat

0.965647

Prob ( F-statistic )

0.000000

APPENDIX 4

Dynamic additive specification: rcons c ryd rcons ( -1 ) ) :

Dependent Variable: RCONS

Method: Least Squares

Date: 10/21/09 Time: 15:34

Sample ( adjusted ) : 1970 2003

Included observations: 34 after accommodations

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

4.506354

5.375165

0.838366

0.4082

RYD

0.202606

0.146306

1.384811

0.1760

RCONS ( -1 )

0.737391

0.179686

4.103786

0.0003

R-squared

0.699289

A A A A Mean dependant volt-ampere

48.39961

0.679888

A A A A S.D. dependant volt-ampere

14.92920

S.E. of arrested development

8.446706

A A A A Akaike info standard

7.189527

Sum squared resid

2211.752

A A A A Schwarz standard

7.324206

Log likeliness

-119.2220

A A A A Hannan-Quinn criter.

7.235457

F-statistic

36.04452

A A A A Durbin-Watson stat

1.424198

Prob ( F-statistic )

0.000000

Dynamic log additive specification: log ( rcons ) c log ( ryd ) log ( rcons ( -1 ) )

Dependent Variable: LOG ( RCONS )

Method: Least Squares

Date: 10/21/09 Time: 15:32

Sample ( adjusted ) : 1970 2003

Included observations: 34 after accommodations

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

0.951086

0.380349

2.500559

0.0179

LOG ( RYD )

0.162741

0.097055

1.676798

0.1036

LOG ( RCONS ( -1 ) )

0.598390

0.164471

3.638278

0.0010

R-squared

0.729374

A A A A Mean dependant volt-ampere

3.837901

0.711914

A A A A S.D. dependant volt-ampere

0.288705

S.E. of arrested development

0.154958

A A A A Akaike info standard

-0.807224

Sum squared resid

0.744374

A A A A Schwarz standard

-0.672545

Log likeliness

16.72281

A A A A Hannan-Quinn criter.

-0.761295

F-statistic

41.77463

A A A A Durbin-Watson stat

1.417667

Prob ( F-statistic )

0.000000

APPENDIX 5

RAMSEY RESET Trial

Dynamic additive specification: rcons c ryd rcons ( -1 ) ) :

Ramsey RESET Test:

F-statistic

2.243951

A A A A Prob. F ( 2,29 )

0.1241

Log likelihood ratio

4.892206

A A A A Prob. Chi-Square ( 2 )

0.0866

Test Equation:

Dependent Variable: RCONS

Method: Least Squares

Date: 12/02/09 Time: 17:47

Sample: 1970 2003

Included observations: 34

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

4.043292

27.65016

0.146230

0.8848

RYD

0.682897

0.530752

1.286660

0.2084

RCONS ( -1 )

1.496027

1.891990

0.790716

0.4355

FITTED^2

-0.044123

0.052895

-0.834168

0.4110

FITTED^3

0.000384

0.000351

1.096192

0.2820

R-squared

0.739589

A A A A Mean dependant volt-ampere

48.39961

0.703670

A A A A S.D. dependant volt-ampere

14.92920

S.E. of arrested development

8.126887

A A A A Akaike info standard

7.163286

Sum squared resid

1915.343

A A A A Schwarz standard

7.387751

Log likeliness

-116.7759

A A A A Hannan-Quinn criter.

7.239835

F-statistic

20.59061

A A A A Durbin-Watson stat

1.752695

Prob ( F-statistic )

0.000000

APPENDIX 6

BREUSCH- GODFERYSERIAL CORRELATION LM Trial

Dynamic additive specification: rcons c ryd rcons ( -1 ) ) :

Breusch-Godfrey Serial Correlation LM Test:

F-statistic

3.260477

A A A A Prob. F ( 2,29 )

0.0528

Obs*R-squared

6.241736

A A A A Prob. Chi-Square ( 2 )

0.0441

Test Equation:

Dependent Variable: Residual oil

Method: Least Squares

Date: 12/02/09 Time: 17:49

Sample: 1970 2003

Included observations: 34

Presample losing value lagged remainders set to zero.

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

6.594438

6.711195

0.982603

0.3339

RYD

0.309798

0.287950

1.075873

0.2909

RCONS ( -1 )

-0.464851

0.391742

-1.186624

0.2450

RESID ( -1 )

0.668505

0.359984

1.857041

0.0735

RESID ( -2 )

-0.116130

0.288298

-0.402812

0.6900

R-squared

0.183580

A A A A Mean dependant volt-ampere

-2.72E-15

0.070971

A A A A S.D. dependant volt-ampere

8.186745

S.E. of arrested development

7.890888

A A A A Akaike info standard

7.104347

Sum squared resid

1805.717

A A A A Schwarz standard

7.328812

Log likeliness

-115.7739

A A A A Hannan-Quinn criter.

7.180896

F-statistic

1.630238

A A A A Durbin-Watson stat

2.130296

Prob ( F-statistic )

0.193398

APPENDIX 7

Dynamic additive specification: rcons c ryd rcons ( -1 ) ) :

Heteroskedasticity Test: White

F-statistic

3.352107

A A A A Prob. F ( 5,28 )

0.0169

Obs*R-squared

12.73126

A A A A Prob. Chi-Square ( 5 )

0.0260

Scaled explained SS

17.71086

A A A A Prob. Chi-Square ( 5 )

0.0033

Test Equation:

Dependent Variable: RESID^2

Method: Least Squares

Date: 12/02/09 Time: 17:52

Sample: 1970 2003

Included observations: 34

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

C

230.9927

246.2392

0.938083

0.3562

RYD

-6.765674

6.695113

-1.010539

0.3209

RYD^2

-0.206819

0.143325

-1.443002

0.1601

RYD*RCONS ( -1 )

0.654743

0.338315

1.935305

0.0631

RCONS ( -1 )

-6.113805

12.64696

-0.483421

0.6326

RCONS ( -1 ) ^2

-0.256126

0.170470

-1.502469

0.1442

R-squared

0.374449

A A A A Mean dependant volt-ampere

65.05153

0.262743

A A A A S.D. dependant volt-ampere

120.7970

S.E. of arrested development

103.7207

A A A A Akaike info standard

12.28007

Sum squared resid

301223.8

A A A A Schwarz standard

12.54942

Log likeliness

-202.7611

A A A A Hannan-Quinn criter.

12.37193

F-statistic

3.352107

A A A A Durbin-Watson stat

2.070800

Prob ( F-statistic )

0.016946

APPENDIX 8

AUGMENTED DICKEY FULLER TEST IN FIRST DIFFERENCES

Null Hypothesis: D ( RCONS ) has a unit root

Exogenous: Changeless

Lag Length: 0 ( Automatic based on SIC, MAXLAG=0 )

t-Statistic

A A Prob. *

Augmented Dickey-Fuller trial statistic

-4.714140

A 0.0006

Test critical values:

1 % degree

-3.646342

5 % degree

-2.954021

10 % degree

-2.615817

*MacKinnon ( 1996 ) nonreversible p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D ( RCONS,2 )

Method: Least Squares

Date: 12/14/09 Time: 13:25

Sample ( adjusted ) : 1971 2003

Included observations: 33 after accommodations

Variable

Coefficient

Std. Mistake

t-Statistic

Prob.A A

D ( RCONS ( -1 ) )

-0.845947

0.179449

-4.714140

0.0000

C

1.811955

1.544072

1.173491

0.2495

R-squared

0.417546

A A A A Mean dependant volt-ampere

0.140490

0.398758

A A A A S.D. dependant volt-ampere

11.13363

S.E. of arrested development

8.632997

A A A A Akaike info standard

7.207752

Sum squared resid

2310.388

A A A A Schwarz standard

7.298450

Log likeliness

-116.9279

A A A A Hannan-Quinn criter.

7.238269

F-statistic

22.22312

A A A A Durbin-Watson stat

1.899706

Prob ( F-statistic )

0.000049

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