**Chapter THREE**

**Methodology**

**3.1 Research Design**

The ultimate intent in this paper was to depict and compare a figure of published theoretical accounts, to supply some comparing of the distributions that result from them, and to find which best suit the Ghanese economic information. This research focuses on the strategic plus allotment theoretical accounts. This is because these theoretical accounts have the inclination to capture several investing series in a individual theoretical account development process. The informations used for the empirical analysis in this paper were taken from the Bank of Ghana informations base. Annually informations were considered because the stochastic plus theoretical accounts used in the survey ( Wilkie, 1986 ; 1995 ; Whitten & A ; Thomas, 1999 ) used similar informations frequence.

The choice of the theoretical accounts is strictly purposive and convenience. Models’ parametric quantities are calculated utilizing the Ghanese economic informations. Subsequently, some statistics are investigated for easy comparing of the theoretical accounts. This helped in placing the best theoretical account for the Ghanese economic information. The theoretical accounts considered were ;

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( a ) The Wilkie theoretical account, as described in Wilkie ( 1995 ) ;

( B ) The ARCH fluctuation of the Wilkie theoretical account, besides described in Wilkie ( 1995 )

( degree Celsius ) The Whitten & A ; Thomas theoretical account, as described in Whitten & A ; Thomas ( 1999 ) .

Before the theoretical accounts comparing, I besides looked at the features of the informations in other to understand and show the nature of the Ghanese economic variables. Statistical univariate clip series analysis were conducted, besides, basic premises for stochastic modeling were checked. Actuarial stochastic modeling normally follow the criterion premise that the theoretical account mistakes are independent and identically distributed ( i.i.d. ) normal random variables and that, in pattern the variables used in the actuarial applications, such as the rising prices or involvement rate, are assumed to be autoregressive and have changeless unconditioned agencies ( Sherris, 1997 ) .

The being of unit roots in the series for theoretical accounts show the nature of the tendencies in the series. If a series contains a unit root so the tendency in the series is stochastic and dazes to the series will be lasting and this can be an accretion of past random dazes otherwise, the series is termed as “trend stationary” . An probe refering the unit root and stationarity of the series were besides conducted. This is because tendency stationary has major deductions for investing theoretical accounts in actuarial applications ( Sherris et al. 1999 ) . The Dickey and Fuller ( 1979 ) trial is employed for this intent.

**3.2 Datas**

Stochastic mold requires the usage of informations from the yesteryear to unite with the present to pattern the hereafter. For a good theoretical account, the construction should be consistent with validated or widely accepted economic and fiscal theory. These theories and the developed theoretical accounts depend on empirical informations for proof. Statistically analyzing the historical informations provides a better penetrations into the characteristics of past experience inherent in the variable that the theoretical account must capture. Good theoretical accounts are consistent with historical informations since the parametric quantity appraisals are normally based on the historical information.

The informations considered were:

- Consumer Price Index ( CPI ) ;
- Ghana Stock Exchange All Share Index ( ASI ) ;
- Share dividend output
- The 90 twenty-four hours bank measure outputs ;
- One twelvemonth note output

Logarithms and differences of the logarithms are used in the analysis of the CPI, ASI, and dividends. The difference in the logarithms of the degree of a series is the continuously compounded tantamount growing rate of the series. The clip series secret plans are used to demo the pictural behaviour of the series used in this research.

**3.3 ANALYTICAL TOOLS**

the descriptive statistics and graph were obtained ( Talk about R )

- Normality Test
- UNIT ROOTS AND STATIONARY SERIES

Stationarity

When we want to gauge a VAR theoretical account, an of import premise is that the historical information is stationary. This means that the belongingss of the procedure such as the mean and the autocovariances are _xed and do non depend on clip T ( purely talking the procedure is covariance stationary under these conditions ) . Stationarity is a important premise for being able to depict the stochastic behaviour of some variable by a individual theoretical account and to be able to gauge the parametric quantities of such a theoretical account on one sample of informations. Otherwise each point in clip would necessitate another theoretical account and merely one observation would be available to gauge

**3.4 THE Model**

The theoretical accounts are explained below demoing the expression and see the nature of the variables as modelled by the several Authors. The formulae specify how each variable is simulated and each theoretical account requires certain parametric quantities. All the theoretical accounts have been calibrated from past informations and the writers have by and large given the values of the parametric quantities from their fitted theoretical account, but by suiting different economic information, it requires re-calibration to deduce the parametric quantity set that are utile to the survey. In order to compare the theoretical accounts in certain respects I shall utilize the same information set for the parametric quantity appraisal.

**3.4.1 THE WILKIE MODEL**

THE WILKIE MODEL

The Wilkie^{’}s theoretical account is a cascade construction embracing assorted investing series. In Wilkie ( 1986, 1995 ) , the rising prices series is assumed to be the impulsive force for the other investing series. The investing series are linked together through a vigorous survey and analysis based on a mixture of statistical grounds and economic premises. The cascade construction of the Wilkie theoretical account and the lineation are given below.

Fig 3.1: Structure of Wilkie theoretical account

Figure 2.1. The Cascade Structure of the Wilkie theoretical account.

The parts of the Wilkie Model development in the UK included four cardinal variables, and these are:

- Retail monetary value index ( Q )
- Share dividends index ( D )
- Dividend output ( Y ) on portion monetary value index ( P )
- Consols output or long–term authorities involvement rate ( C ) .

Each of the variables are modeled within a cascade construction such that they are ordered from the top degree to the lower degrees. The values of the lower degree variables depend on the lagged value of themselves and the values of the variables in the upper degrees ( Chao, 2007 ) . Chao ( 2007 ) explains that the rising prices rate, which is calculated from the alteration of retail monetary value index, depends on its ain lagged values and is placed on the top bed of the construction. The prognosis of this variable depends on its historical grounds. Chao ( 2007 ) describes the dividend output as being in the 2nd bed, and that its anticipation is based on both the historical grounds and that of rising prices rate ; and eventually, the 3rd bed includes dividend and consols, whose prognosis are based on the historical information, that of dividend output, and that of rising prices rate.

The variables of the Wilkie ( 1986, 1995 ) were modeled under the undermentioned processs:

- Measure one, the variables were modeled by the arrested development on the upper degree variables.
- Then, the models’ remainders were tested and constructed through the criterion Box–Jenkins univariate clip series patterning method.
- Next, he tested other methods such as Vector Autoregressive ( VAR ) mold of two correlative variables, and GARCH mold for the discrepancy of remainders.

Formula

**a. The***Price Inflation Model.*

Inflation, as measured by the retail monetary values index ( CPI ) , is modelled by a first order car regressive ( AR ( cubic decimeter ) ) procedure. Wilkie^{’}s AR ( 1 ) monetary value rising prices theoretical account is of the signifier:

Whereis the force of rising prices over twelvemonth ( t-1 ) to ( T ) and it is given as:

Hence:

That isis a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where QMU, QA and QSD are parametric quantities to be estimated.

This theoretical account is described as that, each twelvemonth the force of rising prices is equal to its average rate ( QMU ) , plus a per centum of last twelvemonth ‘s divergence from the mean ( QA ) , plus a random invention which has zero mean and a standard divergence of QSD ( Wilkie, 1986 ) . The premise is that, rising prices, being the factor of economic uncertainness, depends merely on past values of itself. There is important autocorrelation at slowdown 1, which provides statistical justification for inclusion of thevariable, and no other economically plausible autocorrelation or partial autocorrelation is important at 95 % ( Whitten & A ; Thomas, 1995 ) . The AR ( 1 ) theoretical account of the force of rising prices is a statistically stationary series ( i.e. in the long tally the mean and discrepancy are changeless ) , of a

B.*Share Outputs theoretical account*

Share outputs are modelled as a map of the current rising prices rate and the history of their past tendencies. The Wilkie^{’}s AR ( 1 ) theoretical account of the portion dividend output is given as:

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, YMU, YA, YW and YSD are parametric quantities to be estimated.

This theoretical account uses logarithmic transformed dividend output,as the response variable. Wilkie ( 1995 ) described the theoretical account as that, at any day of the month the logarithm of the dividend output is equal to its average value ( ln YMU ) , plus a per centum of its divergence a twelvemonth ago ( YA ) from the mean, plus an extra influence from rising prices ( YW ) times the force of rising prices in the old twelvemonth, plus a random invention which has zero mean and a standard divergence of YSD.

c. The*Dividends theoretical account*

The theoretical account for portion dividends, whereis the value of a dividend index on ordinary portions at clip T, is give as:

Specifyingas the logarithm of the addition in the portion dividends index from twelvemonthto twelvemonth, the Wilkie^{’}s MA ( 1 ) dividend output theoretical account can besides be represented as:

Where,.

Hence, Wilkie ( 1985,1995 ) modeled for P ( T ) , the value of a monetary value index of ordinary portions at clip T as:

Or

.

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, DMU, DB, DW, DX, DSD, DY and DD are parametric quantities to be estimated.

Wilkie ( 195 ) described the theoretical account in words as: “*in each twelvemonth the alteration in the logarithm of the dividend index is equal to a map of current and past values of rising prices, plus a average existent dividend growing ( which is taken as nothing ) , plus an influence from last twelvemonth ‘s dividend output invention, plus an influence from last twelvemonth ‘s dividend invention, plus a random invention which has zero mean and a standard divergence ( DSD )*.”

vitamin D.*Long Term Interest Rate*

The long term involvement rate theoretical account is for the Consols outputThe theoretical account is based on, which isadjusted the long memory consequence of rising prices rate. The Wilkie^{’}s AR ( 1 ) consols yield theoretical account is presented as:

This portion is an exponentially leaden norm of current and past monetary value rising prices, standing the expected hereafter rising prices over the life of the bond.

This is a zero-mean AR ( 1 ) procedure which is independent of monetary value rising prices, and controls the long-run existent involvement rate.

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, CMU, CW, CA, CSD, and Cadmium are parametric quantities to be estimated.

The theoretical account is composed of two parts: an expected hereafter rising prices and a existent output ( Sahin, et tal. ) . The part stand foring the rising prices portion is modelled as a leaden moving mean whiles the existent portion is modelled is an AR ( 1 ) with a part from the dividend output. The parametric quantity,*CW*= 1, which implies that, the theoretical account takes into history, the “Fisher effect” , in which the nominal output on bonds reflects both expected rising prices over the life of the bond and a ’real’ rate of involvement Sahin, et tal. ) . Wilkie ( 1995 ) defines the logarithm of the existent involvement constituentas a additive autoregressive order one or three AR ( 1 ) or AR ( 3 ) but preferred the AR ( 1 ) theoretical account.

vitamin E.*Short Term Interest Rate ( Bank Rate )*

Aside the cardinal parts of the Wilkie theoretical account, I consider one of the subsequent variables modelled by wilkie ( 1995 ) . Wilkie used bank rate or bank base rate series to pattern short-run involvement rates. Short-run involvement rates are clearly connected with long-run 1s. Wilkie’s attack was to pattern the difference between the logarithms of the difference of these series whereis the value of bank rate at clip*T.*

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, BMU, BA, and BSD, are parametric quantities to be estimated.

**3.4.2 THE ARCH MODEL**

**The Wilkie ARCH Model**

The initial theoretical account developed by Wilkie ( 1986 ) assumed that the remainders of the rising prices theoretical account were usually distributed. In 1995, he re-examined his ain theoretical account and observed that the remainders were much fatter tailed than a normal distribution. In Statistics and Econometrics, one of the ways to pattern these fat tailed distributions is utilizing an Autoregressive Conditional Heteroscedastic ( ARCH ) theoretical account ( Engle, 1982 ) . In an Autoregressive Conditional Heteroschedastic ( ARCH ) theoretical account, the discrepancy of the invention term is modelled as a separate procedure ( instead than assumed to be changeless ) ( Wright, 2004 ) . After the re-examination of the historical information, Wilkie ( 1995 ) proposed an ARCH theoretical account for the standard divergence of the rising prices theoretical account.

The ARCH theoretical account was seen to depict the informations better than the original theoretical account by Huber ( 1997 ) and was suggested that, it should by and large be used in applications of the theoretical account, unless the ARCH consequence is non important for those peculiar applications.

In this ARCH model the changing value of the standard divergence, QSD ( T ) , is made to depend on the antecedently observed value of the principal variable, I ( t?1 ) , which itself is modelled by an autoregressive series. The suggested theoretical account ( with a little change in the notation ) was:

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, QMU, QA QSA QSB, and QSC, are parametric quantities to be estimated.

This implies that the fluctuation depends on how far off last year’s rate of rising prices,, was from some in-between degree, QSC ( similar to the mean, QMU ) , but with the divergence squared, so that utmost values of rising prices in either way would increase the discrepancy ( Sahin, 2010 ) .

Comparing the ARCH theoretical account to the initial autoregressive theoretical account showed that, the distribution of the force of monetary value rising prices) exhibits fatter dress suits and a greater concentration around the long-run mean value ( Wright, 2004 ) . The ARCH fluctuation was incorporated in merely the monetary value rising prices theoretical account. Therefore, the balance of the series follow the modeling as in the initial Wilkie theoretical account since Wilkie ( 1995 ) found no basses to re-model them as ARCH theoretical accounts.

The ARCH theoretical account appears to give a better representation of rising prices than the theoretical accounts presuming changeless discrepancy.

**3.4.3 THE WHITTEN AND THOMAS MODEL**

Whitten and Thomas theoretical account

The chief underpinning belief for this theoretical account is that, “the economic system behaves otherwise in times of hyperinflation, than it does in times of normal rising prices levels” Whitten & A ; Thomas ( 1999 ) . This belief is non-linear in nature and hence could non hold been modelled linearly. After vigorous geographic expedition of several alternate, Whitten & A ; Thomas ( 1999 ) adapted the Wilkie theoretical account ( additive theoretical account ) to integrate their non-linearity premise, instead than basically altering the whole preparation.

Whitten & A ; Thomas ( 1999 ) did non pattern the heteroscedastic nature of the monetary value rising prices utilizing the ARCH theoretical account as in Wilkie ( 1995 ) due to the challenges in gauging the theoretical account since and that it can give rise to disturbing consequences from simulation. Whitten & A ; Thoma ( 1999 ) employed the threshold modeling technique since threshold theoretical accounts are besides capable of stand foring conditional discrepancy, and furthermore, exhibit short-run alterations in mean. They proposed two governments for each of the variables. The procedures in each government are similar to those defined by Wilkie ( 1986 ; 1995 ) . Following the same cascade construction above, the expression for the theoretical accounts are given below:

*The Price Inflation Model*

Inflation is assumed to be represented as a SETAR ( self-exciting threshold autoregressive ) theoretical account, with hold 1, and a threshold that differentiates between normal and high rising prices. They fitted many different threshold theoretical accounts. Due to the dearth of informations partitioned into the upper government, it was hard to contend any kind of autocorrelation construction in the hyperinflation government Whiten & A ; Thomas ( 1999 ) . The concluding suitable for threshold theoretical account for the monetary value rising prices is SETAR ( 2 ; 1, 0 ) , therefore:

That isis a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where QMU1, QA1, QSD1, QMU2, and QSD2 are parametric quantities to be estimated.

The theoretical account is described as that if the rising prices in the old twelvemonth was below a certain threshold ( QR ) , so the expected force of rising prices () is equal to its mean ( QMU1 ) , plus a per centum of last year’s divergence from the mean ( QA1 ) plus a random invention which has zero mean and standard divergence QSD1. Conversely if the rising prices in the old twelvemonth was above the threshold, so the expected force of rising prices soon is equal to its mean ( QMU2 ) , plus a random invention which has zero mean and standard divergence QSD2. The theoretical account is able to command heteroscedasticity in a manner because the expected discrepancy of rising prices when it is in its aroused stage is greater than when it is in its quiescent stage Whitten & A ; Thomas ( 1999 ) .

*The dividend theoretical account*

Following Wilkie ( 1986, 1995 ) , Whitten & A ; Thomas ( 1999 ) besides represented the portion divided series as moving norm of order one ( MA ( 1 ) ) . Specifyingas in the Wilkie theoretical account as the logarithm of the addition in the portion dividends index from twelvemonth*T*-1 to*T*, this theoretical account is similar to the Wilkie theoretical account but with the debut of a normal and a high rising prices governments. In economic sense, dividends do better in times of normal rising prices, than in times of high rising prices hence, the theoretical account employs the status that

The theoretical account foris of the signifier:

Where,

That isis a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where Where, DMU1 DMU2, DB, DW, DX, DSD, DY and DD are parametric quantities to be estimated.

The portion output theoretical account

Our portion output theoretical account is different to Wilkie.s ln*Yttrium*(*T*) in that we include a

transportation consequence from ?ln*C*(*T*) to*YN*(*T*) . ln*Yttrium*(*T*) was re-estimated a TAR theoretical account, with excess

parametric quantities*YY*1 and*YY*2, to include this transportation, i.e.

where,

That is,is a series of independent, identically distributed unit normal random variables, ( the premise is that, they have zero average and unit standard divergence ) . Where, YMU, YA, YW and YSD are parametric quantities to be estimated.

consol

It is non easy to gauge the exponential smoothing parametric quantity,*Cadmium*, for each

government in*C*(*T*) . There are jobs when utilizing a more sensitive smoothing parametric quantity, in

that {*C*(*T*) .*Centimeter*(*T*) } & gt ; 0, i.e. we can non let a negative existent involvement rate. It seemed a

necessary simplification to hold the allowance for expected future rising prices over the life

of the bond (*Centimeter*(*T*) ) , and therefore the parametric quantity*Cadmium*, defined the same for each government. It

hence follows that, like the Wilkie theoretical account, our theoretical account gives a unit addition between

rising prices and involvement rates.

4.4.6*C*(*T*) was so re-estimated as a TAR theoretical account, i.e.

short term involvement rate

*Bachelor of divinity*(*T*) was re-estimated as a TAR theoretical account, i.e.

**3.5 COMPARISON OF THE MODELS**

3.6 Parameter appraisal

3.7 Simulation process