Projects prepared for Financial Econometrics II classes at Warsaw School of Economics using GNU R
In this report we're going to:
- analyse the Dutch bond's yield,
- model them using ARIMA and VAR,
- analyse the properties of the estimated models from the econometric point of view,
- calibrate models to obtain 1-period forecasts,
- measure the out-of-sample accuracy,
- compare future forecasts with those published by the EU Commission.
In this chapter we're going to analyse the available data, focusing on the Dutch bonds yield series integration and (partial) autocorrelation.
The above plots are typical for AR(1) (or equivalently infinite MA) process but let's notice that the partial autocorrelation for lag 1 is in fact equal to zero which suggests non-stationarity and clearly makes modelling this series, without any transformation, pointless.
We're going to pursue Phillips-Perron (null hypothesis = non-stationary) and KPSS (null hypothesis = stationary) tests. Dickey-Fuler test would be omitted as the PP test is its extension.
The critical level of significance is usually assumed at 0.05, but in our analysis we're going to treat p-values between 0.01 and 0.1 as not fully conclusive (nfc).
On raw series we've obtained:
- PP test p-value = 0.08239 ~ null (non-stationary, nfc)
- KPSS test p-value = 0.01 ~ alternative (non-stationary)
On first differences series we've obtained:
- PP test p-value = 0.01 ~ alternative (stationary)
- KPSS test p-value = 0.1 ~ null (stationary)
The results suggest that the process is actually integrated at level 1, and the models for the first differences should be used.
ACF for the first differences looks pretty random, with the majority of the lags not being significantly correlated. A partial autocorrelation is significant for levels 1 and 7, and because of that we will set the maximum AR lag at 7 while calibrating the model.
We're now going to calibrate and estimate ARIMA model using the iterative algorithm proposed by Rob Hyndman (2008). We will disable the drift, set the order of differentiation (d) at 1, and the initial p and q values at 7 and 0, as we expect that the ARIMA(7,1,0) would best fits the data. The dataset is enough large (3908 observations), so we will use AIC information criteria instead of AICc.
| p,d,q | aic |
|---|---|
| 7,1,0 | -13888.93 |
| 7,1,1 | -13886.92 |
| 0,1,1 | -13886.66 |
| 1,1,0 | -13886.51 |
| 6,1,1 | -13881.67 |
| 0,1,0 | -13881.29 |
The results met our expectations when it comes to the best model - ARIMA(7,1,0). Unfortunately the differences between the ARIMA(0,1,0), equivalent to random walk, and the other models are very insignificant.
Obtained ARIMA(7,1,0) model has the following specification:
| coefficient | estimate | p-value |
|---|---|---|
| ar1 | 0.0445 | 0.0054 |
| ar2 | -0.0124 | 0.4398 |
| ar3 | -0.0158 | 0.3241 |
| ar4 | 0.0147 | 0.3594 |
| ar5 | -0.0216 | 0.1778 |
| ar6 | 0.0112 | 0.4844 |
| ar7 | -0.0509 | 0.0015 |
and measures:
| measure | value |
|---|---|
| LogLik | 6952.4671 |
| AIC | -13888.9342 |
| BIC | -13838.7700 |
| RMSE | 0.0408 |
Its IRF looks like:
showing that the impulse expires in approximately 3 weeks (15 business days) with the higher variance along first 8 days.
In the further analysis we're going to compare the following ARIMA models:
- 7,1,0
- 1,1,0
- 0,1,0 (RW)
We won't use ARIMA(0,1,1) due to its' property of impulse vanishing after 1 period (day) leaving constant differences same for all horizons of forecast which we assume to be not sufficient solution for long-term prediction.
This sections needs to be further developed. Related code is finished, see analysis.R.