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In this paper, time series modelling is examined with a special application to modelling inflation data in Tanzania. In particular the theory of univariate non linear time series analysis is explored and applied to the inflation data spanning from January 1997 to December 2010. Time series models namely, the autoregressive conditional heteroscedastic (ARCH) (with their extensions to the generalized autoregressive conditional heteroscedasticity ARCH (GARCH)) models are fitted to the data. The stages in the model building namely, identification, estimation and checking have been explored and applied to the data. The best fitting model is selected based on how well the model captures the stochastic variation in the data (goodness of fit). The goodness of fit is assessed through the Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and minimum standard error (MSE). Based on minimum AIC and BIC values, the best fit GARCH models tend to be GARCH(1,1) and GARCH(1,2). After estimation of the parameters of selected models, a series of diagnostic and forecast accuracy test are performed. Having satisfied with all the model assumptions, GARCH(1,1) model is found to be the best model for forecasting. Based on the selected model, twelve months inflation rates of Tanzania are forecasted in sample period (that is from January 2010 to December 2010). From the results, it is observed that the forecasted series are close to the actual data series.

The concept of time series is based on the historical observations. It involves explaining past observations in order to try to predict those in the future [

Inflation as described by [

In recent years, inflation has become one of the major economic challenges facing most countries in the world especially those in Africa including Tanzania. Inflation is a major focus of economic policy worldwide as de- scribed by [

The most common way of measuring inflation is the consumer price index (CPI) over monthly, quarterly or yearly. The inflation rate

where

Within time series modelling, there are two approaches available for forecasting: the univariate and multiva- riate. In particular this paper will forecast future values of inflation time series data using the univariate fore- casting approach, in which forecasts depend only on present and past values of a single series being forecasted.

Let

where

and the error term

with non-negativity condition that

The ARCH (1) model is a particular case of the general ARCH

with non-negativity condition that

The ARCH models provide good estimates of the series before it is realized. Let

minimum mean squared error prediction, that is

of the observations. Then according to [

The forecasts for the

[

The obvious possible problem in using the ARCH formulation is that the approach can lead to a highly para- metric model if the

A process

where

with the restrictions

The GARCH

Let

where

In order to find the GARCH

Substituting Equation (12) into the Equation (11) one gets

Therefore

Multiplying both sides of Equation (14) by

But

and since

Thus the autocovariance of the squared returns for the GARCH

Dividing both sides of Equation (15) by

Letting

By Equation (17),

Assuming

with

The l-step ahead forecast of the conditional variance in a GARCH

where

This section is dedicated to fitting the GARCH family of models to the Tanzania inflation rate data which we obtained from the Tanzania National Bureau of Statistics. The original data set consist of 168 monthly observa- tions of the Tanzania inflation rates spanning from January 1997 to December 2010 as shown in

To avoid the difficulties of possible premature convergence we perform a pre-fit analysis. This will lead to selecting the appropriate model that adequately describes the data. In this pre-estimation or pre-fit analysis, data are loaded in the form of a price series, and then converted to a return series (stabilized series). The pre-fit anal- ysis checks the return series for correlation and then quantifies the correlation. Because GARCH modelling as- sumes a return series, we need to convert inflation data (raw data) to returns.

The returns appear to be quite stable over time and the transformations from Inflation rate data to returns has produced a stationary time series. The GARCH model assumes that return series is a stationary process. This may seem limiting, but the inflation data to return transformation is common and generally guarantees a stable data set for GARCH modelling.

According to [

In

Statistical test for heteroscedasticity is carried out in order to establish the presence of ARCH effects in the data. This is shown in

. Summary for statistics for Tanzania’s monthly inflation

Period | Average | Standard Deviation |
---|---|---|

Jan 1997-Sept 2001 | 9.94 | 4.3 |

Oct 2001-July 2005 | 5.1 | 1.58 |

Aug 2005-April 2009 | 8.24 | 2.52 |

May 2009-Dec 2010 | 8.9 | 3.0 |

Overall Period | ||

Jan 1997-Dec 2010 | 8.1 | 3.7 |

Time plot of monthly inflation in Tanzania

First difference of Log of CPI

ACF with bounds for raw return series

PACF with bounds for the raw returns series

ACF of the squared returns

. Ljung-Box-Pierce Q-test for autocorrelation (at 95% confidence)

Lag | H | p-value | Stat | Critical Value |
---|---|---|---|---|

10 | 0 | 0.0721 | 17.1019 | 18.3070 |

15 | 0 | 0.2359 | 18.529 | 24.9958 |

20 | 0 | 0.3888 | 21.1416 | 31.4104 |

. Ljung-Box-Pierce Q-test for squared returns (at 95% confidence)

Lag | H | P-value | Stat | Critical Value |
---|---|---|---|---|

10 | 1.000 | 0.0000 | 57.3782 | 18.3070 |

15 | 1.000 | 0.0000 | 76.7057 | 24.9958 |

20 | 1.000 | 0.0000 | 82.7525 | 31.4104 |

. Engle ARCH test for heteroscedasticity (at 95% confidence)

Lag | H | p-value | Stat | Critical value |
---|---|---|---|---|

10 | 1.000 | 0.0000 | 68.6467 | 18.3070 |

15 | 1.000 | 0.0000 | 67.9727 | 24.9958 |

20 | 1.000 | 0.0000 | 66.2913 | 31.4104 |

From

The strategy used in selecting the appropriate model from competing models is based on the Akaike information criterion

MATLAB software is used to perform trial and error evaluations to determine the best fitting model. The idea is to have a parsimonious model that captures as much variation in the data as possible. Usually the simple GARCH model captures most of the variability in most stabilized series. Small lags for p and q are common in applications. Typically GARCH

From the derived models, using the method of maximum likelihood the estimated parameters of GARCH

The standard errors are used to assess the accuracy of the estimates, the smaller the better. The model fit sta- tistics used to assess how well the model fit the data are the AIC and BIC. The corresponding values are: AIC = 474.8 and BIC = 487.3 with the log likelihood function of 233.4. The standard errors are quiet small suggesting precise estimates. Based on 95% confidence level, the coefficients of the GARCH

One of the assumptions of GARCH models is that, for a good model, the residuals must follow a white noise process.

It can be observed that both innovations and returns exhibit volatility clustering. However if we plot the, standardized innovations (the innovations divided by their conditional standard deviation), they appear generally stable with little clustering as seen in

The time plot of the residuals given in

. Parameter estimates for GARCH(1,1)

Parameter | C | K | GARCH (1) | ARCH(1) |
---|---|---|---|---|

Estimates | 0.0272 | 0.0753 | 0.4573 | 0.5427 |

Standard Error | 0.0283 | 0.0254 | 0.0681 | 0.0958 |

t-value | 0.9627 | 2.9656 | 6.7136 | 5.6619 |

0.8315 | <0.0001 | <0.0001 | <0.0001 |

. Comparison of suggested GARCH models

Model | AIC | BIC | MSE | Log-Likelihood |
---|---|---|---|---|

GARCH(0,1) | 489.0200 | 498.3560 | 0.1026 | 241.5100 |

GARCH(1,1) | 474.8236 | 487.2715 | 0.0544 | 233.4118 |

GARCH(0,2) | 478.5589 | 491.0068 | 0.0860 | 235.2794 |

GARCH(1,2) | 491.1419 | 491.1419 | 0.0863 | 232.7910 |

GARCH(2,1) | 476.8236 | 492.3835 | 0.0735 | 233.4118 |

GARCH(2,2) | 477.1047 | 495.7766 | 0.0654 | 232.5524 |

Plot for return, estimated volatility and innovations (residuals)

Time plot of residuals from GARCH(1,1)

Histogram of residuals from GARCH (1, 1)

the histogram of the residuals from the GARCH(1,1) model. The histogram shows almost a symmetric bell- shaped distribution which is indicative of the residuals following a normal distribution. The slight negative skewness is expected since the residuals may also come from student’s

. The first two forecast error statistics depend on the scale of the dependent variable. These are used as relative measure to compare forecasts for the same series across different models. The smaller the error, the better the fore

. Forecast Accuracy Test on the most likely suggested GARCH models

Model | MSE | MAE | RMSE | Thiele’s U test |
---|---|---|---|---|

GARCH(1,1) | 0.5848 | 0.7483 | 0.8651 | 0.8528 |

GARCH(1,2) | 0.6034 | 0.9812 | 0.7768 | 0.9821 |

. Inflation forecast by GARCH(1,1) model for period of January 2010 to December 2010

Month | Forecast (%) | Observed value (%) | Forecast error |
---|---|---|---|

January | 11.32 | 10.9 | 0.42 |

February | 10.43 | 9.6 | 0.83 |

March | 9.66 | 9.0 | 0.66 |

April | 10.08 | 9.4 | 0.66 |

May | 7.90 | 7.2 | 0.70 |

June | 8.74 | 7.9 | 0.84 |

July | 7.29 | 6.3 | 0.99 |

August | 7.48 | 6.6 | 0.88 |

September | 5.03 | 4.5 | 0.53 |

October | 4.81 | 3.9 | 0.91 |

November | 5.14 | 4.3 | 0.84 |

December | 6.30 | 5.6 | 0.70 |

casting ability of that model. The remaining two statistics are scale invariant. The Theil inequality coefficient always lies between zero and one, where zero indicates a perfect fit.

From

The

It can be observed from the

In this paper, time series modelling was examined with a special application to modelling inflation data in Tan- zania. In particular, the theory of univariate nonlinear time series analysis was explored and applied to the infla- tion data spanning from January 1997 to December 2010. The best fitting model was selected based on how well the model captures the stochastic variation in the data. Based on minimum Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) values, it was observed that the best fit GARCH models were GARCH(1,1) and GARCH(1,2). However, after estimation of the parameters of selected models, a series of diagnostic and forecast accuracy test were performed and GARCH(1,1) model was found to be the best. Based on the selected model, twelve months inflation rates of Tanzania were forecasted in sample period (from January 2010 to December 2010). From the results, it is observed that the forecasted series are close to the actual data series.