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Time Series Analysis

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Stephanie phelan

on 15 April 2014

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Transcript of Time Series Analysis

Time series Analysis
Annual prices of natural gas from 1990-2007
Nonlinear Trend
Trend Component
Quarterly starts of single-family houses from 2006-2008
Linear Trend
Time Series Analysis
The study of data generated over time
Consists of 4 major components
Trend (T)
Cyclical (C)
Seasonal (S)
Irregular (I)
The long-term behavior of the time series
Types of trends:
Linear trend
Non-linear trend
No-trend
No Trend
By: Stephanie Phelan
Linearly decreasing overall trend
Quadratically increasing overall trend
Annual closing price for Abbot Laboratories stock
The data is fluctuating around the mean of 46.4 indicating no trend
Cyclical Component
Fluctuations of the data around the trend over a period of time longer than a year.
Examples:
Stock Prices
Unemployment Rate
Gross National Product
Cyclical Component
The number of people in labor force who are unemployed
1990-1995, 2008-2013 : data fluctuating above the trend
1995-2008: data fluctuating under trend
Seasonal Component
Fluctuations in data over a single year that can be attributed to seasonal changes
Examples:
Toy stores sales highest in December
Firework sales highest in June-July
Snow shovel sales lowest during summer months
Seasonal Component
Quarterly Umbrella Sales displays strong seasonal effect
Highest number of sales in Quarter 2 because of higher rainfall in the months of May-August

Lowest number of sales in Quarter 4 because of low rainfall in September - December
Irregular Component
Index Numbers
Index numbers measures the change in a variable over time, relative to the variable's original value at the base period.
Types of indexes:
Simple index
Simple Composite index
Weighted Index
Simple Index
Steps to calculate:
Gather the prices for a single commodity over a period of time
Select a base period
Calculate the index for each period using the formula:

Simple Index Example
Table 8.1 displays price and production of three types of metals in the U.S.

Simple Index Continued
Compute simple index for each metal using January as base period
Copper price index for Jan. :
Copper price index for Dec. :
Percentage change in copper prices is 158.9-100= 58.9%. This indicates a 58.9% price increase for copper from January to December
Simple Composite Index
An index that measures the price of two or more commodities
Steps to calculate :

Gather the prices of the two or more commodities over a period of time
Select a base period
Use the formula to compute the index:
An index that measures a price of a single commodity
where :
is the price at period t
is the price at the base period
Simple Composite Index Example
Compute the composite price index for the three metals in Table 8.1
Composite price index for Jan. :
Composite price index for Dec. :
Percentage change in the total prices = 144.1-100= 44.41%. This indicates a 44.41% increase in the total price of the three metals
Weighted Composite Indexes
A weighted composite index accounts for different quantity levels among the commodities
Types of weighted indexes:
Laspeyres Index
Paasche Index
Laspeyres Index
Steps to calculate:
First collect the prices for each period. Denote the prices using:
Select the base period:
Third we collect the quantities for each period . Denote the quantities using :
Lastly, calculate the index at period t using the formula:
A Laspeyres index uses the base period quantities to weight the index.
Laspeyres Index
Compute the Laspeyres index using price and production data of the three metals in Table 8.1, January is used as the base period
Laspeyres Index for Jan. :
Laspeyres Index for Dec. :
The percentage change in price of the three metals is 121.88-100= 21.88%
Where:
Irregular component is fluctuations in the data caused by rare events or human actions
Examples:
Drought causes crop prices to drastically increase
Natural disasters increase death toll and damage costs
Bankruptcy of a company causes low stock prices (Enron)
Or random variations due to human actions
Paasche Index
The Paasche index uses the current period quantities to weight the index
Steps to Calculate:
First, collect the price for each period. Denote the prices using:
Second, select the base period:
Third, collect the quantities for each period. Denote the quantities using :
Lastly, compute the index at time period t using the formula:
Paasche Index
Compute the Paasche index using the price and production data of the three metals in Table 8.1 January is used as the base period
Paasche index for Jan. :
Paasche index for Dec.:
The percentage change in price of the three metals is 122.11-100=22.11%
Forecasting method

Moving average method
purpose is to remove irregular fluctuations
Used to find the underlying trend
Forecasts the next value/s of the time series
High level of accuracy for short-range forecasts
Moving Averages Method
Uses the average of the previous k values of the time series in order to forecast the next value of the series
Steps to calculate:
Determine you span length: k
Retrieve the most recent k values
Use the formula to calculate the forecast for the next period: t+1:
Where:
is the forecast of the time series for period t+1
is the actual value at time period t
Multiplicative model:
is the time series value at time period t
Moving Averages Example
Table 12.1 displays the median sales price in thousands for new home sales in the month of April for years 1990-2011
Moving Average Example
We compute moving average using k=3,5,7 to compare the effect of k length on forecasting the median new home sales price for April 2011
2011 forecast k=3:
2011 forecast k=5:
2011 forecast k=7:
Moving Average Example
Moving average of length three,224.63, is the most accurate of the actual 2011 value, 224.7
Moving Average Plot
Examples of Laspeyres Index:
Consumer Price
Producer Price
Dow Jones

Conclusion
Times series contain many components and in order to make accurate forecasts we must be able to predict the behavior of the series.
I will use these techniques to decompose a time series of strawberry prices in my poster presentation.
We accomplish this through identifying the different components of the series, the use of index numbers, and the use of moving averages
Comparison
Simple Composite Index : 44.41% increase

Laspeyres Index : 21.88% increase
Paasche Index: 22.11% increase
Percentage change between January and December for the total price of Copper, Steel, and Lead
Mentor: Dr. Dagys
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