A case study with stock market data
Modeling Time-series Stochastic Data
VECTOR auto-regressive (VAR) integrated model comprises multiple time series and is quite a useful tool for forecasting. It can be considered an extension of the auto-regressive (AR part of ARIMA) model. VAR model involves multiple independent variables and therefore has more than one equations. Each equation uses as its explanatory variables lags of all the variables and likely a deterministic trend. Time series models for VAR are usually based on applying VAR to stationary series with first differences to original series and because of that, there is always a possibility of loss of information about the relationship among integrated series.
Therefore, differencing the series to make them stationary is one solution, but at the cost of ignoring possibly important (“long run”) relationships between the levels. A better solution is to test whether the levels regressions are trustworthy (“cointegration”.) The usual approach is to use Johansen’s method for testing whether or not cointegration exists. If the answer is “yes” then a vector error correction model (VECM), which combines levels and differences, can be estimated instead of a VAR in levels. So, we shall check if VECM is been able to outperform VAR for the series we have.
This an extension of my previously published article.
Loading all datasets ( Gold, Silver & Crude Oil)
After necessary cleaning & pre-processing (filling the missing values with previous ones), we finally have the three time series for necessary analysis.
A quick test is to check if the data is random. Random data will not exhibit a structure in the lag plot.
The time series plot clearly indicates some kind of relationships among the series. The linear shape of the lag plot suggests that an AR model is a better choice. We also don’t see any outlier in the data. Data here showing linear pattern, indicating the presence of positive auto-correlation.
“Time series for economic data is generally stochastic or has a trend that is not stationary, meaning that the data has a root unit”
# plots the autocorrelation plots at 75 lags
for i in dataset:
plot_acf(dataset[i], lags = 50)
plt.title(‘ACF for %s’ % i)
result = adfuller(time_series.values)
print('ADF Statistic: %f' % result)
print('p-value: %f' % result)
for key, value in result.items():
print('\t%s: %.3f' % (key, value))
Through the above function, we can run Augmented Dickey Fuller (ADF) test on all columns, which clearly shows the original series are non-stationary and contain unit root.
print('Augmented Dickey-Fuller Test: Gold Price Time Series')
print('Augmented Dickey-Fuller Test: Silver Price Time Series')
d Dickey-Fuller Test: Oil Price Time Series')
VAR models can also be used for analyzing the relation between the variables involved using Granger Causality tests. Granger causality speciﬁes that a variable y1t is causal for a variable y2t if the information in y1t is helpful for improving the forecasts of y2t.
Granger Causality tests try to determine if one variable(x1) can be used as a predictor of another variable(x2) where the past values of that another variable may or may not help. This means that x1 explains beyond the past values of x2. Two important assumptions here are -
- both x1 and x2 are stationary
- there exists a linear relation between their current and past values.
This means that if x1 and x2 are non-stationary, we have to make them stationary before testing for Granger Causality.
Split the Series into Training and Testing Data
We will fit the VAR model on X_train to forecast the next 10 observations. These forecasts will be compared against the actuals present in test data (X_test). We shall use multiple forecast accuracy metrics.
A difference transform is a simple way for removing a systematic structure from the time series. We will remove trend by subtracting the previous value from each value in the series which is the first order differencing. To keep it simple, we will do first order differencing or seasonal differencing.
If we have an integrated order to n time series and if we take first order to difference and time, we will be left with series integrated order of zero.
X_train_log = np.log(X_train)
looking at the plot, we could figure out that data set looks like normalized
Auto-Correlation Function Analysis of Transformed Series
Below function plots the auto-correlation plots for the difference in each stock’s price from the price the previous trading day at 75 lags.
fig, ax = plt.subplots(1,2, figsize=(10,5))
ax = plot_acf(X_train_log_diff['Gold'], ax=ax)
ax = plot_pacf(X_train_log_diff['Gold'], ax=ax)
We have shown ACF & PACF of transformed Gold series; likewise, other series can be plotted.
ADF Test - transformed series
print('Augmented Dickey-Fuller Test: Gold Price Time Series')
print('Augmented Dickey-Fuller Test: Silver Price Time Series')
print('Augmented Dickey-Fuller Test: Oil Price Time Series')
Augmented Dickey-Fuller Test on "Oil"
The Granger causality test is conducted to determine whether one time series is useful in forecasting another. A time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y.
Here, for multivariate Granger causality analysis performed by fitting a VAR to the time series. Considering below is a d-dimensional multivariate time series —
Granger causality is performed by fitting a VAR model with L time lags as follows:
where ε ( t ) is a white Gaussian random vector, and A τ is a matrix for every τ. A time series X i is called a Granger cause of another time series X j, if at least one of the elements A τ ( j , i ) for τ = 1 , … , L is significantly larger than zero.
print(grangercausalitytests(X_train_log_diff[['Gold','Silver']], maxlag=15, addconst=True, verbose=True))
print(grangercausalitytests(X_train_log_diff[['Gold','Oil']], maxlag=15, addconst=True, verbose=True))
print(grangercausalitytests(X_train_log_diff[['Oil','Silver']], maxlag=15, addconst=True, verbose=True))
Below output shown for Gold & Oil which differs the test hypothesis till lag 4.
A VAR(p) process in its basic form is:
Here yt represents a set of variables collected in a vector, c denotes a vector of constants, a is a matrix of autoregressive coefficients and et is white noise. Since the parameters of a are unknown, we have to estimate these parameters. Each variable in the model has one equation. The current (time t) observation of each variable depends on its own lagged values as well as on the lagged values of each other variable in the VAR.
I have implemented Akaike’s Information Criteria (AIC) through the VAR (p) to determine the lag order value. In the fit function, I have passed a maximum number of lags and the order criterion to use for order selection.
#Initiate VAR model
model = VAR(endog=X_train_log_diff)
res = model.select_order(15)
#Fit to a VAR model
model_fit = model.fit(maxlags=3)
#Print a summary of the model results
Forecast VAR Model
The forecasts are generated on the training data used by the model.
# Get the lag order
lag_order = model_fit.k_ar
print(lag_order)# Input data for forecasting
input_data = X_train_log_diff.values[-lag_order:]
pred = model_fit.forecast(y=input_data, steps=nobs)
pred = (pd.DataFrame(pred, index=X_test.index, columns=X_test.columns + '_pred'))
So, to bring it back up to its original scale, we need to de-difference to the original input data. Our data is 1st logarithm transformed and then differenced. So, to inverse, we have to first use cumulative sum to de-differentiate and then use exponential. Natural logarithm is the inverse of the exp().
# inverting transformation
def invert_transformation(X_train, pred_df):
forecast = pred.copy()
columns = X_train.columns
for col in columns:
forecast[str(col)+'_pred'] = X_train[col].iloc[-1] + forecast[str(col) +'_pred'].cumsum()
return forecastoutput = invert_transformation(X_train, pred)
print(output)output_original = np.exp(output)
VAR Forecast evaluation
#Calculate forecast bias
forecast_errors = [X_test['Oil'][i]- output_original['Oil_pred'][i] for i in range(len(X_test['Oil']))]
bias = sum(forecast_errors) * 1.0/len(X_test['Oil'])
print('Bias: %f' % bias)#Calculate mean absolute error
mae = mean_absolute_error(X_test['Oil'],output_original['Oil_pred'])
print('MAE: %f' % mae)#Calculate mean squared error and root mean squared error
mse = mean_squared_error(X_test['Oil'], output_original['Oil_pred'])
print('MSE: %f' % mse)
rmse = sqrt(mse)
print('RMSE: %f' % rmse)
“Least squares parameter estimation of dynamic regression models is known to exhibit substantial bias in small samples when the data is fairly persistent”
VECM imposes additional restriction due to the existence of non-stationary but co-integrated data forms. It utilizes the co-integration restriction information into its specifications. After the cointegration is known then the next test process is done by using error correction method. Through VECM we can interpret long term and short term equations. We need to determine the number of co-integrating relationships. The advantage of VECM over VAR is that the resulting VAR from VECM representation has more efficient coefficient estimates.
In order to fit a VECM model, we need to determine the number of co-integrating relationships using a VEC rank test.
vec_rank1 = vecm.select_coint_rank(X_train, det_order = 1, k_ar_diff = 1, method = 'trace', signif=0.01)
We find the λtrace statistics in the third column, together with the corresponding critical values. The test statistic of 38.25 is lower than the critical value (41.08) and so the null of at most one co-integrating vector cannot be rejected.
Let us employ an alternative statistic, the maximum-eigenvalue statistic (λmax).
vec_rank2 = vecm.select_coint_rank(X_train, det_order = 1, k_ar_diff = 1, method = 'maxeig', signif=0.01)
The test output reports the results for the λmax statistics which does not differ much from trace statistic; the critical value (29.28) is still higher than test statistic.
We will still go ahead and estimate VECM, since it can still valuable for short-run dynamics in absence of co-integration. Let’s estimates the VECM on the prices with 9 lags, 1 co-integrating relationship, and a constant within the co-integration relationship. I have used ‘cili’ a combination of “ci” — constant within the co-integration relation and “li” — linear trend within the co-integration relation
vecm = VECM(endog = X_train, k_ar_diff = 9, coint_rank = 3, deterministic = ‘ci’)
vecm_fit = vecm.fit()
forecast, lower, upper = vecm_fit.predict(10, 0.05)
print(“lower bounds of confidence intervals:”)
print(“\nupper bounds of confidence intervals:”)
VECM Forecast evaluation
Though we had an indication that, VAR would be best for our data set for price prediction; however, we have shown VECM for experimentation and illustration purpose. This is a simple procedure to explain VAR. However, there are other procedures like impulse response analysis and variance decomposition also can be introduced to experiment if we are able to see how a shock to one variable affects other variable in subsequent periods.
Time series for economic data is generally stochastic or has a trend that is not stationary, meaning that the data has a root unit. To be able to estimate a model using the data-
Steps for VAR-
- Test stationarity of data and degree of integration
- Determination of lag length
- Test the granger causality
- Estimation of VAR
- Variance decomposition
Forecasting Steps for VECM-
- Determination of lag length
- Test the granger causality
- Cointegration degree test
- Estimation of VECM
- Variance decomposition
I can be reached here.
Notice: The programs described here are experimental and should be used with caution. All such use at your own risk.
(1) Rao, B. (2007). Cointegration: for the Applied Economist, Springer.
(2) Ashley, R. A., & Verbrugge, R. J. (2009). To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models. International Journal of Data Analysis Techniques and Strategies, 1(3), 242–274.
(3) Lütkepohl, H. (2011). Vector autoregressive models. In International Encyclopedia of Statistical Science (pp. 1645–1647). Springer Berlin Heidelberg.
(4) Kuo, C. Y. (2016). Does the vector error correction model perform better than others in forecasting stock price? An application of residual income svaluation theory. Economic Modelling, 52, 772–789.
- Examine the Data.
- Test for stationarity. 2.1 If the data is non-stationary, take the difference. ...
- Train Test Split.
- Grid search for order P.
- Apply the VAR model with order P.
- Forecast on new data.
- If necessary, invert the earlier transformation.
Stock price modeling in this research is using multivariate time series analysis that is VAR (Vector Autoregressive) and VECM (Vector Error Correction Modeling). VAR and VECM models not only predict more than one variable but also can see the interrelations between variables with each other.
The vector autoregressive (VAR) model is a workhouse multivariate time series model that relates current observations of a variable with past observations of itself and past observations of other variables in the system.
Re: When to use VAR/VECM
You should use VECM if 1) your variables are nonstationary and 2) you find a common trend between the variables (cointegration).
The Vector Error Correction Model (VECM)
If a set of variables are found to have one or more cointegrating vectors then a suitable estimation technique is a. VECM (Vector Error Correction Model) which adjusts to both short run changes in variables and deviations from. equilibrium.
- Analyze the time series characteristics.
- Test for causation amongst the time series.
- Test for stationarity.
- Transform the series to make it stationary, if needed.
- Find optimal order (p)
- Prepare training and test datasets.
- Train the model.
(EViews10): Estimate and Interpret VECM (1) #var #vecm ... - YouTube
(EViews10): Estimate and Interpret VECM (2) #var #vecm ... - YouTube
What's the difference between an error correction model (ECM) and a Vector Error correction model (VECM)? Are these arguments right? -An error correction model is a single equation. A VECM is a multiple equation model based on a restricted VAR.
How Do You Calculate Value at Risk? There are three ways to calculate VAR: the historical method, the variance-covariance method, and the Monte Carlo method. The historical method examines data from prior observations, with the assumption that future results will be similar.
VAR models (vector autoregressive models) are used for multivariate time series. The structure is that each variable is a linear function of past lags of itself and past lags of the other variables.
normally it is the case in VAR modeling, getting a big sample is not a big issue. Zehra Dogan Caliskan is right if you fit a full VAR(1) model with a constant vector because you will have 5 parameters for the constant vector and 5² = 25 parameters for the VAR(1) coefficient.
Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregressive model by allowing for multivariate time series.
A Python variable is a symbolic name that is a reference or pointer to an object. Once an object is assigned to a variable, you can refer to the object by that name. But the data itself is still contained within the object. For example: >>> >>> n = 300.
Multiple Time Series Forecasting With Scikit-Learn - YouTube
Introduction Intuition behind VAR Model Formula Building a VAR model in Python Import the datasets Visualize the Time Series Testing Causation using Granger’s Causality Test Cointegration Test Split the Series into Training and Testing Data Check for Stationarity and Make the Time Series Stationary How to Select the Order (P) of VAR model Train the VAR Model of Selected Order(p) Check for Serial Correlation of Residuals (Errors) using Durbin Watson Statistic How to Forecast VAR model using statsmodels Train the VAR Model of Selected Order(p) Invert the transformation to get the real forecast Plot of Forecast vs Actuals Evaluate the Forecasts Conclusion. You need at least two time series (variables) The time series should influence each other.. It is considered as an Autoregressive model because, each variable (Time Series) is modeled as a function of the past values, that is the predictors are nothing but the lags (time delayed value) of the series.. Intuition behind VAR Model formula How to check the bi-directional relationship using Granger Causality Procedure to building a VAR model in Python How to determine the right order of VAR model Interpreting the results of VAR model How to generate forecasts to original scale of time series. Since you have multiple time series that influence each other, it is modeled as a system of equations with one equation per variable (time series).. In order to forecast, the VAR model expects up to the lag order number of observations from the past data.
Univariate versus Multivariate Time Series Univariate Time Series Multivariate Time Series. Since this article will be focused on multivariate time series, I would suggest you go through the following articles which serve as a good introduction to univariate time series:. But I’ll give you a quick refresher of what a univariate time series is, before going into the details of a multivariate time series.. A univariate time series, as the name suggests, is a series with a single time-dependent variable.. A Multivariate time series has more than one time-dependent variable.. A series like this would fall under the category of multivariate time series.. Now that we understand what a multivariate time series looks like, let us understand how can we use it to build a forecast.. In this section, I will introduce you to one of the most commonly used methods for multivariate time series forecasting – Vector Auto Regression (VAR) .. In a VAR model, each variable is a linear function of the past values of itself and the past values of all the other variables.. We need to forecast the value of these two variables at time t, from the given data for past n values.. Since the AR process is used for univariate time series data, the future values are linear combinations of their own past values only.. In this case, we have only one variable – y, a constant term – a, an error term – e, and a coefficient – w. In order to accommodate the multiple variable terms in each equation for VAR, we will use vectors.. From the above equations (1) and (2), it is clear that each variable is using the past values of every variable to make the predictions.. Similar to the Augmented Dickey-Fuller test for univariate series, we have Johansen’s test for checking the stationarity of any multivariate time series data.. Creating a validation set for time series problems is tricky because we have to take into account the time component.
Visualizing a Time Series Patterns in a Time Series Additive and multiplicative Time Series How to decompose a Time Series into its components?. Stationary and non-stationary Time Series How to make a Time Series stationary?. Series TimeseriesNote, in the series, the ‘value’ column is placed higher than date to imply that it is a series.. You can do a classical decomposition of a time series by considering the series as an additive or multiplicative combination of the base level, trend, seasonal index and the residual.. A stationary series is one where the values of the series is not a function of time.. Differencing the Series (once or more) Take the log of the series Take the nth root of the series Combination of the above. Random White Noise Detrending a time series is to remove the trend component from a time series.. Deseasonalize Time Series The common way is to plot the series and check for repeatable patterns in fixed time intervals.. Secondly, when it comes to time series, you should typically NOT replace missing values with the mean of the series, especially if the series is not stationary.. For example, a random time series with fewer data points can have a lower ‘approximate entropy’ than a more ‘regular’ time series, whereas, a longer random time series will have a higher ‘approximate entropy’.
Hello everyone, In this tutorial, we’ll be discussing Time Series Analysis in Python which enables us to forecast the future of data using the past data that is collected at regular intervals of time.. Let us see the Components of Time Series.. So these are the components of a Time Series data.. It is important that the Series is stationary because working and operating upon Stationary data is much simpler.. We see that the ADF value is -1.22 it is greater than all threshold values of 0.10, 0.05, 0.01 Therefore our Time-Series data is Non-Stationary and it same result as we are getting using the Summary statistics or Histogram Plots.. During tests for stationarity if we have found that our time-series is stationary then we are not required to do any transformation but if we are confirmed that our time-series is non-stationary like in the dataset we are working with we have to perform the transformations.. Removing Trend & Seasonality We have to decompose our data_array separately into Trend, Seasonal and Residual components because we may require to check their values to see that if there are null values or not as they may require to be removed.. Just take the Log of the data and apply ADF Test to see the result.. This model can be fitted to time series data in order to forecast or predict future data in the time- series.. This model can also be used even if the time series is not stationary.. We have to get understandings of the plots and other statistics to find the right order of difference.. Try to change them To make forecasts just split the data into training and testing sets, then fit data the model using training data and then make forecast() method on the test data and just compare the test data with the predicted data.
In this example the model got 80 samples correctly predicted over the population of 100 persons, in this case the accuracy for this model is 80% (80/100).. What if, for the same population, we are predicting who will win the lottery?If the model just predict “No” all the time the accuracy would be 99,9%!. Precision is the ability of the classifier not to label as positive a sample that is negative.. These 10 persons are called True Positive , the model predicted they would go to the beach and they did it!. Supposing the model labeled all 100 persons as positive (voted candidate “A”) the recall would be 100% because the model found all the positive cases.. Confusion matrix is a well detailed metric that brings True Positive (TP), False Negative (FN), False Positive (FP) and True Negative (TN) values.. In this article I covered the most used evaluation metrics for binary classification models!
How to Model Volatility with ARCH and GARCH for Time Series Forecasting in Python - Machine Learning Mastery ›
The problem with variance in a time series and the need for ARCH and GARCH models.. # create a simple white noise with increasing variance. from random import gauss. from random import seed. from matplotlib import pyplot. # seed pseudorandom number generator. seed(1). # create dataset. data = [gauss(0, i*0.01) for i in range(0,100)]. # plot. pyplot.plot(data). pyplot.show(). # check correlations of squared observations. from random import gauss. from random import seed. from matplotlib import pyplot. from statsmodels.graphics.tsaplots import plot_acf. # seed pseudorandom number generator. seed(1). # create dataset. data = [gauss(0, i*0.01) for i in range(0,100)]. # square the dataset. squared_data = [x**2 for x in data]. # create acf plot. plot_acf(squared_data). pyplot.show(). # example of ARCH model. from random import gauss. from random import seed. from matplotlib import pyplot. from arch import arch_model. # seed pseudorandom number generator. seed(1). # create dataset. data = [gauss(0, i*0.01) for i in range(0,100)]. # split into train/test. n_test = 10. train, test = data[:-n_test], data[-n_test:]. # define model. model = arch_model(train, mean='Zero', vol='ARCH', p=15). # fit model. model_fit = model.fit(). # forecast the test set. yhat = model_fit.forecast(horizon=n_test). # plot the actual variance. var = [i*0.01 for i in range(0,100)]. pyplot.plot(var[-n_test:]). # plot forecast variance. pyplot.plot(yhat.variance.values[-1, :]). pyplot.show(). The dataset may not be a good fit for a GARCH model given the linearly increasing variance, nevertheless, the complete example is listed below.. # example of ARCH model. from random import gauss. from random import seed. from matplotlib import pyplot. from arch import arch_model. # seed pseudorandom number generator. seed(1). # create dataset. data = [gauss(0, i*0.01) for i in range(0,100)]. # split into train/test. n_test = 10. train, test = data[:-n_test], data[-n_test:]. # define model. model = arch_model(train, mean='Zero', vol='GARCH', p=15, q=15). # fit model. model_fit = model.fit(). # forecast the test set. yhat = model_fit.forecast(horizon=n_test). # plot the actual variance. var = [i*0.01 for i in range(0,100)]. pyplot.plot(var[-n_test:]). # plot forecast variance. pyplot.plot(yhat.variance.values[-1, :]). pyplot.show()