Money, Exchange Rate, Prices and Output in Nigeria: A Test of the P-Star Model

The search for robust model to predict inflation within a QTM framework gave birth to P-star model which has attracted less attention of researchers and practitioners in Nigeria. This study applied the methodology to high frequency Nigerian data from 1995M1 to 2018M6 to determine the validity of the model for Nigeria using error correction model (ECM). The result supports the working of the model but with slight modification. The modification centres on the incorporation of foreign price gap, (open economy view of inflation), reserve money (Friedmanic/monetarist view), price per litre of petroleum motor spirit (PMS) and output gap (Structuralist view). With this modification, P-star model proved to be a viable inflation forecasting alternative model for Nigeria. Consequently, the Central Bank of Nigeria is advised to consider adopting this modified version of the model to forecast inflation for Nigeria at least as a complimentary model to be used side-by-side with the existing forecasting model of the Bank. This will no doubt enhance the efficacy of the monetary policy of the Bank as such policies will be predicated on sufficient information, particularly on the future path of inflation.

Where M is money stock, V represents the velocity of transaction, P is the price level and Y denotes aggregate income. The equation, at any given period, equates the money stock (M) times its velocity (V) to total income (Y) generated in the economy times their prices (P). Fisher (1935) assumes that velocity and output are constant in the short-run and that flexible wages and prices are necessary conditions for the attainment of full employment.
With preference for P, equation 1 is transformed as:

=
(2) If we denote the equilibrium price level consistent with money supply as P e and with the knowledge that P e relates to long-run potential or equilibrium output as well as long-run velocity otherwise referred to as equilibrium velocity, we have a remodeled equation 2 as: The superscript -e -denotes the long-run levels otherwise referred to as equilibrium.
If we divide equation 3 by equation 2 we have: Redefining the variables using lower cases as the natural logs yields: From equation 5, it follows that (p e -p) is decomposed into velocity and output gaps as (v e -v) and (y ey), respectively.
For price gap to correspond to inflation, error correction model can be employed to consolidate both the long-run mechanisms and short-run dynamics. This yield: Subject to: The objective is principally to minimise lnY* Where Y is the variable of interest, Y* is the trend otherwise known as the potential, Y t-1 is a period lag of the variable and ƛ is the Lagrange multiplier which control the smoothness of the variance of the data. It is given as 100, 1600 and 14400 for annual, quarterly and monthly series, respectively.

Statistical Properties of the Data
The characteristics of the data used for the estimation was first explored using descriptive statistics.
From Table 1 which displays the results, it is evident that there are one hundred (100) observations per variable. The table also shows that most of the data are positively skewed except for lrm, lry and ry_gap which are log of reserve money, log of real output and output gap, respectively. Augmented Dickey Fuller (ADF) based on Akaike Information Criterion (AIC) and Phillips-Perron (PP) unit root tests were adopted to characterize the long-run properties of the variables. As reported in Table 2, the result reveals that consumer price index (CPI) and its log version (lcpi), exchange rate (exr), reserve money (rm) and the price of petroleum motor spirit (PMS) are first difference stationary I(1) based on both methodologies, hence enter the regression at first difference, while domestic price gap (lcpi e -lcpi), foreign price gap (exr e -exr), real output (ry) and real output gap (ry e -ry) are stationary at level I(0) and hence treated as such.

Forecasting Performance
In line with Kiptui (2013), the forecasting performances of the models (i.e., Models 1 to 5) are evaluated using five criteria, namely: root mean square error (RMSE), mean absolute error (MAE), Theil inequality, bias and variance proportions. Table 4 presents the results of the evaluation criteria across all models. The results returned Model 5 as the best model implying that Model 5 outperforms all others. By implication, this validates the applicability of P-star model for Nigeria but with slight modification.
Using the domestic price gap alone as contained in the first column of Table 1 cannot predict movement of prices in Nigeria. More so, the inclusion of foreign price gap improves the model but not as good as when other variables such as reserve money, PMS and output gap are included in the model. combined. It is important to note that the forecast is carried out using both static and dynamic models but only the results of the static model are presented here, because they seem to outperform those of the dynamic models.
Close scrutiny of the figures show that all models forecast traced the actual inflation closely but Figure   5 representing Model 5 traced the actual far more closely, further supporting the position of Table 4.

Conclusion and Policy Advice
The importance of the possibility of predicting the future path of inflation to proactive monetary policy formulation cannot be overemphasised. This is so, considering that price stability is said to be the The search for robust model to predict inflation in a QTM framework gave birth to P-star model which attracts less attention of researchers in Nigeria. This study applies the methodology to high frequency Nigerian data from 1995M1 to 2018M6 to test the validity of the model for Nigeria. The result supports the working of the model but with slight modification by considering foreign price gap to capture the import dependent nature of the Nigeria economy. Other considerations include the inclusion of reserve money, price per litre of PMS and output gap.
With this modification, P-star model can be said to be a viable inflation forecasting alternative model for Nigeria. Consequently, the monetary authority, in this case, Central Bank of Nigeria, is hereby advised to consider adopting this modified version of the model to forecast inflation for Nigeria at least as a complimentary model to be used side-by-side with the existing forecasting model of the Bank. This, if done will enhance the efficiency of the monetary policy of the Bank as policies will be predicated on sufficient information, particularly on the future path of inflation.