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**