Fundamental analysis-based approach

Chapter 4: Research Methodology

4.2 Fundamental analysis-based approach

4.2.1 Data Collection for Fundamental Analysis

I have fully adhered to the guidelines and principles governing ethical research conduct in New Zealand, and Unitec Research Ethics Committee (UREC) declared that I do not require ethics approval. This decision is made because all the data used in this thesis are obtained from public databases, and no data about human participants are used in my thesis. The time series of the NZX 50 Index and four macroeconomic variables are collated from several online public databases, and more details are presented in Section 5.3 in Chapter 5.

Most macroeconomic and financial time-series data are non-stationary and must be converted into stationary data before they are used for analysis and modelling. For a stationary time series, the mean, variance and autocorrelation structure should be constant over time.

Logarithmic transformation and differencing are commonly used data conversions for stationary. The order of integration of a time series symbolized by I(d) is the summary statistic representing the minimum number of differences required to convert a non-stationary time series to a stationary process. A time series is said to be integrated of order one [I(1)] if the first difference of that time series is stationary (that is integrated of order zero [I(0)]. Further evidence about the selection, the data sources of the variables, and the data transformation are presented in Chapter 5.

4.2.2 Stationarity, Cointegration, VECM, and Granger Causality Tests

Most economic and financial time series appeared to be non-stationary [243]–[247]. Thus, it is necessary to test the stationarity of the series before investigating any causal relationships.

Augmented Dickey-Fuller unit root test (ADF, [248], [249]) and Phillips-Perron (PP) unit root test [250] are carried out as two formal tests for stationarity. Adhering to the recommendation by Enders [251], we use both unit root tests as the analysis, reinforcing and strengthening the

test decisions. Akaike Information Criterion (AIC, [134], [135]) and Schwarz Information Criterion (SIC, [136]) will be used to verify the optimal number of lags for both unit root tests.

It is apparent that multivariate economic and financial time series appear to be nonstationary; however, the overall behaviour of the multivariate time series may be modelled by a stationary process when an appropriate transformation is applied to the time series ([243]–

[247] et al.). A series of successive differences (𝑑) of a non-stationary time series can be converted to a stationary time series, and these differences are denoted by 𝐼(𝑑) where 𝑑 is the order of integration. If two or more time series are integrated of an order 𝑑 [𝐼(𝑑)] but a linear combination of them depicts a lower order of integration, then these series are said to be cointegrated, as cointegration is based on a long-run relationship between the variables. If cointegration exists, then the errors arising from the bivariate or multivariate regression should be stationary [error term should be 𝐼(0)] [251].

A cointegration test is performed to ascertain whether there is a correlation between multiple time series in the long run. Gonzalo & Lee [252] highlighted that Johansen's [69]–

[71] cointegration test and Engle-Granger's [63] cointegration test are based on two different rationales. Johansen's cointegration test process is based on maximizing correlations (canonical correlation), while Engle-Granger minimizes variances (in the spirit of principal components) [247]. Thus, [252] recommended using both tests as a robustness check. Therefore, the Engle- Granger [63] and Johansen’s cointegration [69]–[71] tests are used in this analysis. Unit root tests (ADF & PP) and cointegration tests (Johansen & Engle-Granger) are well-known standardised tests; thus, they are considered common knowledge and will not be explained in this thesis.

When the non-stationary time series are cointegrated, the long-run equilibrium relationship can be represented as the cointegrating equation. However, [63] highlighted that

although the cointegrating equation accommodates the long-run equilibrium relationship, there could be deviations from the equilibrium in the short run. To overcome this issue, [63] proposed incorporating one period of lagged residuals from the vectors to generate a Vector Error Correction Model (VECM) as specified in Eq. (4.1) – Eq. (4.5).

The VECM requires the non-stationary multivariate time series to be cointegrated; thus, the model enables to perform the GC test (Granger, [30]).

𝛥 𝑙𝑛 𝑦_𝑡 = 𝜇₁+ ∑ 𝛽_11,𝑖

𝑝

𝑖=1

𝛥 𝑙𝑛 𝑦_𝑡−𝑖+ ∑ 𝛽_12,𝑖

𝑝

𝑖=1

𝛥 𝑙𝑛 𝐼𝑁𝐹_𝑡−𝑖+ ∑ 𝛽_13,𝑖

𝑝

𝑖=1

𝛥 𝑙𝑛 𝐸𝑋𝐶_𝑡−𝑖

+ ∑ 𝛽_14,𝑖

𝑝

𝑖=1

𝛥 𝑙𝑛 𝐼𝑁𝑇_𝑡−𝑖 + ∑ 𝛽_15,𝑖

𝑝

𝑖=1

𝛥 𝑙𝑛 𝑆&𝑃_𝑡−𝑖+ 𝜓₁𝐸𝑇𝐶_𝑡−1+ 𝜀_𝑡^𝑁𝑍𝑋 (4.1)

Δ ln 𝐼𝑁𝐹_𝑡 = 𝜇₂+ ∑ 𝛽_21,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝑦_𝑡−𝑖+ ∑ 𝛽_22,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝐹_𝑡−𝑖+ ∑ 𝛽_23,𝑖

𝑝

𝑖=1

Δ ln 𝐸𝑋𝐶_𝑡−𝑖+ ∑ 𝛽_24,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝑇_𝑡−𝑖

+ ∑ 𝛽_25,𝑖

𝑝

𝑖=1

Δ ln 𝑆&𝑃_𝑡−𝑖 + 𝜓₂𝐸𝑇𝐶_𝑡−1+ 𝜀_𝑡^𝐼𝑁𝐹 (4.2)

Δ ln 𝐸𝑋𝐶_𝑡= 𝜇₃+ ∑ 𝛽_31,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝑦_𝑡−𝑖+ ∑ 𝛽_32,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝐹_𝑡−𝑖+ ∑ 𝛽_33,𝑖

𝑝

𝑖=1

Δ ln 𝐸𝑋𝐶_𝑡−𝑖+ ∑ 𝛽_34,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝑇_𝑡−𝑖

+ ∑ 𝛽_35,𝑖

𝑝

𝑖=1

Δ ln 𝑆&𝑃_𝑡−𝑖 + 𝜓₃𝐸𝑇𝐶_𝑡−1+ 𝜀_𝑡^𝐸𝑋𝐶 (4.3)

Δ ln 𝐼𝑁𝑇_𝑡= 𝜇₄+ ∑ 𝛽_41,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝑦_𝑡−𝑖+ ∑ 𝛽_42,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝐼𝑁𝐹_𝑡−𝑖+ ∑ 𝛽_43,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝐸𝑋𝐶_𝑡−𝑖+ ∑ 𝛽_44,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝐼𝑁𝑇_𝑡−𝑖

+ ∑ 𝛽45,𝑖 𝑝

𝑖=1

Δ 𝑙𝑛 𝑆&𝑃𝑡−𝑖 + 𝜓4𝐸𝑇𝐶𝑡−1+ 𝜀_𝑡^𝐼𝑁𝑇 (4.4)

Δ ln 𝑆&𝑃_𝑡= 𝜇₅+ ∑ 𝛽_51,𝑖

𝑝

𝑖=1

Δ 𝑙𝑛 𝑦_𝑡−𝑖+ ∑ 𝛽_52,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝐹_𝑡−𝑖+ ∑ 𝛽_53,𝑖

𝑝

𝑖=1

Δ ln 𝐸𝑋𝐶_𝑡−𝑖+ ∑ 𝛽_54,𝑖

𝑝

𝑖=1

Δ ln 𝐼𝑁𝑇_𝑡−𝑖

+ ∑ 𝛽_55,𝑖

𝑝

𝑖=1

Δ ln 𝑆&𝑃_𝑡−𝑖 + 𝜓₅𝐸𝑇𝐶_𝑡−1+ 𝜀_𝑡^𝑆&𝑃 (4.5)

where 𝑙𝑛 𝑦_𝑡 denotes the natural logarithm of the NZX 50 Index in period 𝑡, 𝑙𝑛 𝐼𝑁𝐹_𝑡 stands for the natural logarithm of New Zealand inflation in period 𝑡, 𝑙𝑛 𝐸𝑋𝐶_𝑡 represents the natural logarithms of the New Zealand exchange rate in period 𝑡, 𝑙𝑛 𝐼𝑁𝑇_𝑡 symbolises the natural logarithms of New Zealand interest rate in period 𝑡, 𝑙𝑛 𝑆&𝑃_𝑡 denotes the natural logarithms of the S&P 500 Index in period 𝑡, Δ refers to the difference operator, 𝑡 is the time period, 𝑝′𝑠 refer to the number of lags, 𝐸𝑇𝐶_𝑡−1 is the error correction term derived from the long-run cointegration relationship, 𝜀_𝑡′𝑠 are the serially uncorrelated error terms, and 𝜇′𝑠, 𝛽′𝑠, and 𝜓′𝑠 are the parameters to be estimated.

We examine the Granger Causality (GC) between NZX 50 Index and the macroeconomic variables used in this study ([28]–[31]) and [253]). [30] established that if all the time series variables are integrated of order one and are cointegrated, there should be GC in at least one direction as one or some variables can forecast the other. This means that, even if the t-statistic (p-value) of the error correction term is statistically insignificant, as long as the F statistic is statistically significant, it is possible to conclude that GC still exists through the short-run effects. Thus, the F statistic, which is calculated by estimating Eq. (4.1) – Eq. (4.5) in both unconstrained and constrained forms, is performed, and it is summarised in Eq. (4.6).

𝐹𝑐=

(𝐸𝑆𝑆𝑅 − 𝐸𝑆𝑆𝑈) 𝑛

( 𝐸𝑆𝑆𝑈

𝑇 − 𝑚 − 𝑛)

(4.6)

Where ESSR is the residual sum of squares of the restricted regression; ESSU is the residual sum of squares of the unrestricted regression; T is the number of observations in the unrestricted regression; m and n are the optimal order of lags.