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Homework 4: Time Series (25pts)

Task 1: Loading the data – 5 pts

In this part, you will load the data from the file volume_per_year.csv. There are dates

and market volume across the years. You can load this file into the variable volume.

print (volume.head())

Month volume

0 1949-01 22400

1 1949-02 23600

2 1949-03 26400

3 1949-04 25800

4 1949-05 24200

Be aware, when you load a file, the dates are loaded as strings. You will need to use

read_cvs wisely.

Task 2: Stationarity – 5 pts

A common assumption in many time series techniques is that the data are stationary.

A stationary process has the property that the mean, variance and autocorrelation

structure do not change over time.

Questions:

A- Plot the volume across the years.

B- What do you deduce from the plot?

C- Testing stationarity

To test stationarity, we can use the Dickey-Fuller test or Rolling statistics (such as Moving

Average and Moving variance)

Step1: Calculate the moving average with a window of 1 year. Store into a variable ma

Step2: Calculate the moving standard deviation with a window of 1 year. Store into a

variable msd

Step3: Plot on the same graph:

Volume (blue), ma (green) and msd (red)

Step4: What do you conclude?

Step5: Using the package from statsmodels.tsa.stattools import adfuller

You will confirm your conclusion of the Step4 by finding this ouput:

In [21]: print (adtestoutput)

Test Statistic 0.815369

p-value 0.991880

#Lags Used 13.000000

Number of Observations Used 130.000000

Critical Value (1%) -3.481682

Critical Value (10%) -2.578770

Critical Value (5%) -2.884042

What is the null hypothesis of the Dickey-Fuller test?

What do you conclude?

Task 3: Make a Time Series stationary – 5pts

If the time series is not stationary, we can often transform it to stationarity with one of

the following techniques.

1- We can difference the data.

That is, given the series Zt, we create the new series

Yi=Zi?Zi?1.

The differenced data will contain one less point than the original data. Although you can

difference the data more than once, one difference is usually sufficient.

2- For non-constant variance, taking the logarithm or square root of the series may

stabilize the variance. For negative data, you can add a suitable constant to make

all the data positive before applying the transformation. This constant can then be

subtracted from the model to obtain predicted (i.e., the fitted) values and forecasts

for future points.

Questions:

A- We are going to try to eliminate the trend previously observed.

Plot the logarithm of the volume.

B- What do you observe?

C- We are now going to try to smooth the data.

Store the logarithm of the volume data into a variable logvolume

Store the moving average with a 1-year window into a variable mavolume

Plot the graph representing logvolume and mavolume.

The red shows the trend. You just need to subtract logvolume – mavolume and store it

into volume_without_trend.

D- Retest stationarity the same way as you did in the task 2.

E- Redo the study with an exponentially weighted movin average with a half period of

one year. pd.ewma(your_data,halflife=12)

F- Retest stationarity for ewma. G- What do you

conclude with this different method

Task 4: Removing trend and seasonality with differencing – 5pts

Questions:

A- Remove the stationarity apply differencing to the log volume data.

You will need to use the function shift.

B- Plot the graph

C- Test stationarity

Task5: Forecast Time Series – 5pts

https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average

ARIMA (Auto-Regressive Integrated Moving Averages) forecasting for a stationary time

series is a linear regression equation.

Predictors depend on the parameters (p Number of AR terms ,d Number of

Differences,q Number of MA terms) of the ARIMA model.

A- You need to study the ACF

Use the package:

from statsmodels.tsa.stattools import acf, pacf

Calculate the acf of the diff log volume obtained in the previous section.

Be aware of removing the non-defined values. If you don’t do that, acf will return NAs. You

can use the function dropna to remove these undefined values.

plt.subplot(121)

plt.plot(acf)

plt.axhline(y=0,linestyle='--',color='gray')

plt.axhline(y=-1.96/np.sqrt(len(volume_log_diff)),linestyle='--',color='gray')

plt.axhline(y=1.96/np.sqrt(len(volume _log_diff)),linestyle='--',color='gray')

plt.title('Autocorrelation Function')

B- Finally you will load the library

from statsmodels.tsa.arima _model import ARIMA

C- You will run the ARIMA model using p=2, d=1, q=2 on the log date (not

differentiated since d=1). You can store the result of this function into the

variable model

D- You will store the result of model.fit(disp=-1) into results_ARIMA

E- You will plot the log volume with results_ARIMA.

F- We need to convert the predicted values into the original scale one

predictions_ARIMA_diff = pd.Series(results_ARIMA.fittedvalues, copy=True)

print (predictions_ARIMA_diff.head())

Month

1949-02-01 0.009580

1949-03-01 0.017491

1949-04-01 0.027670

1949-05-01 -0.004521

1949-06-01 -0.023890

Find the function converting diff values into real one. (you should be able to use cumsum)

predictions_ARIMA_diff_cumsum.head())

Month

1949-02-01 0.009580

1949-03-01 0.027071

1949-04-01 0.054742

1949-05-01 0.050221

1949-06-01 0.026331

G- Apply exponential to go back to the initial scale