Econometric issues

5.2 Data analysis

5.2.5 Econometric issues

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where 𝐶𝐿𝑇𝐴_𝑖𝑡 is current liabilities to total assets, 𝑃_{𝐺𝑅𝑂𝑊𝑇𝐻𝑖𝑡}and 𝑁_{𝐺𝑅𝑂𝑊𝑇𝐻𝑖𝑡} represent positive and negative sales growth respectively, 𝑆𝐼𝑍𝐸_𝑖𝑡 is the natural logarithm of market capitalisation, a proxy for firm size, 𝐼_𝑖𝑡 is fixed investment during the year t deflated by total assets, 𝑂𝐶𝐹𝑇𝐴_𝑖𝑡 is operating cash flows to total assets; 𝐿𝐸𝑉𝐸𝑅𝐴𝐺𝐸_𝑖𝑡 is the amount of debt employed by the firm and is deflated by total assets; 𝑅𝐺𝐷𝑃_𝑖𝑡 is the Real GDP growth rate, 𝑀𝐾𝑇𝑃𝑂𝑊𝐸𝑅_𝑖𝑡 is the market power of the firm and 𝜂_𝑖 represents unobservable heterogeneity, 𝜆_𝑡 are the time dummy variables and 𝜀_𝑖𝑡 is the error term.

The study repeated the estimation of Equation (8) using the disaggregated approach the working capital finance sources, 𝐶𝐿𝑇𝐴_𝑖𝑡 comprising accounts payable, short-term debt and accruals.

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the error term. This correlation does not disappear when 𝑁 in the sample gets larger or 𝑇 increases (Bond, 2002).

GMM in first differences was considered superior to the alternative approach of estimating Equation (5) the fixed-effects model, the least-squares dummy variables (LSDV). Although the Fixed Effects estimator, eliminates the firm-specific effects 𝜂_𝑖 through the within transformation, it does not eliminate bias. The transformed lagged dependent variable 𝑦_{𝑖,𝑡−1}− 𝑦_𝑖.−1 will still be correlated with (𝜀_𝑖𝑡− 𝜀̅_𝑖.); where 𝑦_𝑖.−1 = ∑^𝑇_𝑖=2^{𝑦𝑖.−1}_𝑇 − 1, even if the random disturbances are not serially correlated because 𝑦_{𝑖,𝑡−1} is correlated with 𝜀̅_𝑖. by construction. 𝜀_𝑖𝑡 is correlated with 𝑦̅_𝑖. because the latter contains 𝑦_𝑖𝑡. The fixed effect estimator produces biased but consistent estimates when 𝑇 tends to infinity and not when 𝑁 tends to infinity. This is known as the dynamic panel bias or the Nickell bias (Nickell, 1981). While the LSDV fails to deal with the problem of 𝐸(∆𝑦_{𝑖,𝑡−1}∆𝜀_𝑖,𝑡) ≠ 0, Generalised Method of Moments takes care of this problem by using the lagged dependent variable (𝑦_{𝑖,𝑡−𝑠} 𝑓𝑜𝑟 𝑠 ≥ 2) in level as instruments.

Using OLS regression on first differenced equations produces biased and inconsistent estimates of the parameters because (𝑦_{𝑖,𝑡−1}− 𝑦_{𝑖,𝑡−2}) and (𝜀_𝑖𝑡− 𝜀_{𝑖,𝑡−1}) are correlated through the terms 𝑦_{𝑖,𝑡−1} and 𝜀_{𝑖,𝑡−1}. The fixed effects estimator fails to produce consistent estimates when 𝑁 tends to infinity and 𝑇 is fixed. GMM in first differences produces consistent estimates because it was designed for 𝑁 tends to infinity and 𝑇 is fixed; that is, small-T and large-N panels.

The Instrumental Variable (IV) estimator as suggested by Anderson and Hsiao (1981), produces consistent and efficient estimates in a dynamic panel. The IV estimator takes the first differenced equation and finds a set of variables, the instruments, to apply the instrumental variable estimator. Instruments are used to eliminate the correlation between the regressors and the disturbances because they must be correlated with the regressors but uncorrelated with the disturbances. In this case, since (𝑦_{𝑖,𝑡−1}− 𝑦_{𝑖,𝑡−2}) and (𝜀_𝑖𝑡− 𝜀_{𝑖,𝑡−1}) are correlated, (𝑦_{𝑖,𝑡−2}) or (𝑦_{𝑖,𝑡−2}− 𝑦_{𝑖,𝑡−3}) are used as an instrument for (𝑦_{𝑖,𝑡−1}− 𝑦_{𝑖,𝑡−2}) because they are

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uncorrelated with (𝜀_𝑖𝑡− 𝜀_{𝑖,𝑡−1}) but correlated with (𝑦_{𝑖,𝑡−1}− 𝑦_{𝑖,𝑡−2}). Anderson and Hsiao (1982) suggest that as long the error terms are not serially correlated 𝑦_{𝑖,𝑡−2} is the obvious choice for an instrument for (𝑦_{𝑖,𝑡−1}− 𝑦_{𝑖,𝑡−2}).

The Anderson and Hsiao (1981) estimator (henceforth termed the AH estimator) when the dimension of a panel is (𝑁 × 𝑇) can be written as

𝛿_𝐴𝐻 = (𝑍¹𝑋)⁻¹𝑍¹𝑌 ) … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 10

where 𝑍 is a 𝐾 × 𝑁 (𝑇 – 2) matrix of instruments, 𝑋 is a 𝐾 × 𝑁 (𝑇 – 2) of regressors and 𝑌 is a 𝑁 (𝑇 – 2) × 1 vector of dependent variables.

𝑍 = [

𝑌_𝑖,1 ∆𝑥_𝑖,3

. .

. .

𝑌_{𝑖,𝑇−2} ∆𝑥_𝑖,𝑇

] 𝑋 = [

∆𝑌_𝑖,2 ∆𝑥_𝑖,3

. .

. .

∆𝑌_{𝑖,𝑇−1} ∆𝑥_𝑖,

] 𝑌 = [

∆𝑌_𝑖,3 . .

∆𝑌_𝑖,𝑇

] … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 11

𝑍 = [ 𝑍₁

. . 𝑍_𝑁

] 𝑋 = [ 𝑋₁

. . 𝑋_𝑁

] 𝑌 = [ 𝑌₁

. . 𝑌_𝑁

] … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 12

The IV estimation produces consistent estimates if the error term in levels is not serially correlated. However, its weakness is that it fails to use all the available moments, which means that it does not necessarily result in more efficient estimates.

GMM in first differences as advanced by Arellano and Bond (1991) produces more efficient and consistent estimates, hence its preference over the AH estimator. It deploys additional instruments obtained by applying the moment conditions that exist between the lagged dependent variable and the disturbances. The number of moment conditions depends on 𝑇, the time periods, which are derived from the first differenced equation. Generalised Method of Moments uses the lagged dependent variables plus the lagged values of exogeneous regressors

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as instruments and a weighting matrix which takes into account the moving averages (MA) (1) process in the differenced error term and the general heteroscedasticity. As a result, the Generalised Method of Moments estimates result in smaller variances than those associated with the AH type instrumental variable estimators.

The Generalised Method of Moments estimator can be expressed as follows 𝜃̂_𝐺𝑀𝑀 = (𝑋¹𝑍^∗𝐴_𝑁𝑍^∗1𝑋)⁻¹𝑋¹𝑍^∗𝐴_𝑁𝑍^∗1𝑌 … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 13

Where 𝜃 ̂ is vector of coefficient estimates of exogeneous and endogeneous regressors, 𝑋̅ and 𝑦̅ are the vectors of first differenced regressors and dependent variables, respectively, 𝑍 is a vector of instruments and 𝐴_𝑁 is a vector used to weight the instruments.

𝑋 = [ 𝑋₁

. . 𝑋_𝑁

] 𝑌 = [ 𝑌₁

. . 𝑌_𝑁

] … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 14

Generalised Method of Moments uses an instrument matrix of the form,

𝑍_𝑖 = [

[𝑦_𝑖0, ∆𝑥₂^′ 0 ⋯ 0

0 [𝑦_𝑖0, 𝑦_𝑖2, ∆𝑥₃^′] ⋱ 0

⋮ 0

0 0 0 [𝑦_𝑖0, … , 𝑦_𝑖𝑇−2∆𝑥_𝑇^′]

… … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 15

where the rows correspond to the first differenced equation for the period 𝑡 = 3, 4, … , 𝑇 for individual 𝑖 and exploit the moment conditions,

𝐸[𝑍_𝑖^′∆𝜀_𝑖] = 0 … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 16 for 𝑖 = 1, 2 , … , 𝑁 where ∆𝜀_𝑖 = (∆𝜀_𝑖3, ∆𝜀_𝑖4, … , ∆𝜀_𝑖𝑇) ′.

Arellano and Bond (1991) proposed two estimators; the one-step estimator and the two-step estimator (henceforth termed GMM1 and GMM2, respectively). GMM2 is the optimal estimator. GMM1 turns out to be optimal when the residuals are homoscedastic. If there is heteroscedasticity, GMM1 of instrumental variables continues to be consistent; however,

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carrying the estimation in two steps increases efficiency. The weight matrix of a GMM1 is given by;

𝐴_1𝑁 = (1

𝑁∑ 𝑍_𝑖^∗′

𝑁

𝑖=1

𝐻𝑍_𝑖^∗) −¹… … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 17

where H is a T – 2 square matrix with twos in the main diagonals, minus ones in the first sub diagonals, and zeros otherwise.

The weight matrix of a GMM2 if given by, 𝐴_𝑁 = (1

𝑁∑ 𝑍_𝑖^∗′

𝑁

𝑖

∆𝜀̂∆𝜀_𝑖 ̂_𝑖^′𝑍_𝑖^∗) −¹ … … … … 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 18

Where ∆𝜀̂_𝑖 = ∆𝜀̂_𝑖… , ∆𝜀̂_𝑖𝑇 are the residuals from a consistent GMM1 of ∆𝑦_𝑖.

5.3 WORKING CAPITAL AND FIRM VALUE RELATIONSHIP ESTIMATION MODEL