Time Series Analysis: Predicting Housing Index in the United States

Project Objective

The goal of this project is to conduct a Time Series Analysis on the United States Housing Price Index (HPI) using data from the Federal Housing Finance Agency (FHFA). The objective is to analyze historical trends, apply statistical modeling techniques, and forecast future housing prices

Dataset

Source : Federal Housing Finance Agency (FHFA)
Time Period : January 1991 - April 2013
Sampling Frequency : Monthly observations
Number of Observations Used : First 80 samples (January 1991 - July 1997) for model building

Objectives

1. Identify a suitable time series model for forecasting the U.S. Housing Price Index
2. Generate predictions for the next five years based on the selected model

Methodology

Data Preprocessing

• The dataset was preprocessed to remove unnecessary attribute
• Stationarity was checked using Augmented Dickey-Fuller (ADF) test
• Differencing was applied to achieve stationarity
• Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots were analyzed to determine model order

Model Selection

Two models were considered from the EACF analysis:

1. ARI(1,1) Model (AutoRegressive Integrated Model)
2. IMA(1,1) Model (Integrated Moving Average Model)

• Parameter estimation was performed using Maximum Likelihood Estimation (MLE)
• Model assumptions (independence, normality, adequacy) were validated through
  • Shapiro-Wilk Test (Normality Check)
  • Runs Test (Independence Check)
  • Ljung-Box Test (Model Adequacy Check)

Model Comparison

Model	Parameter Estimate	Standard Error	AIC	Normality	Independence	Adequacy
ARI(1,1)	0.6486	0.0858	37.89	✅ Pass	✅ Pass	✅ Pass
IMA(1,1)	0.5070	0.0879	53.24	✅ Pass	✅ Pass	❌ Fail

• Since ARI(1,1) passed all diagnostics, it was selected as the final model

Forecasting

• The ARI(1,1) model was used to predict HPI for the next 10 months (September 1997 - June 1998)
• Prediction intervals were calculated for each forecasted value

Forecasted Results

Date	Predicted HPI	Lower PI	Upper PI
1997-09-01	120.959	120.366	121.553
1997-10-01	121.128	119.984	122.271
1997-11-01	121.237	119.559	122.915
1997-12-01	121.308	119.129	123.486
1998-01-01	121.354	118.712	123.995
1998-02-01	121.383	118.314	124.452
1998-03-01	121.403	117.939	124.867
1998-04-01	121.415	117.585	125.246
1998-05-01	121.423	117.251	125.596
1998-06-01	121.429	116.936	125.921

Key Findings & Insights

• The ARI(1,1) model effectively captures the trend in the U.S. Housing Price Index

• The predicted values align closely with actual observed values

• Forecast uncertainty increases over time, as indicated by the widening prediction intervals

• The model suggests a gradual increase in housing prices, assuming economic conditions remain constant

Limitations

• The model was trained on only 80 data points, limiting its predictive power for long-term forecasting.

• External factors such as economic policies, inflation, mortgage rates, and housing supply were not included in the model.

• Future improvements can include more advanced models (e.g., ARIMA, LSTM) and external economic indicators for better accuracy

Technologies Used

• Programming Language: R

• Libraries: forecast, tseries, stats

• Statistical Methods: ADF Test, Maximum Likelihood Estimation, Ljung-Box Test

References

1. Case, K. E., & Shiller, R. J. (2003). Is there a bubble in the housing market? Brookings Papers on Economic Activity.

2. Green, R., & Hendershott, P. H. (1996). Age, housing demand, and real house prices. Regional Science and Urban Economics.

3. Rapach, D. E., & Strauss, J. K. (2009). Differences in housing price forecastability across U.S. states. International Journal of Forecasting.

4. Olick, D. (2021). Home prices in July see the biggest jump in 17 years. CNBC.

5. U.S. Bureau of Economic Analysis. Table 2.4.5U: Personal Consumption Expenditure