Approaching an analysis of various factors on house prices

I've been given a set of data of 506 observations for 14 variables. One of which is house prices in a particular area, and the other 13 are all factors which could (potentially) have an effect on those house prices, such as per capita crime rate by town, air qualities, tax rates, etc.

My goal is to try and fit an appropriate model to the data, which predicts the house price from the other variables, which I'm guessing should involve the least squares regression. I need help to try and figure out which direction I should take in order to write up a report/analysis on this data, as I'm inexperienced with compiling reports on databases this large.

My main question really is: What methods of approach should I take in order to perform a statistical analysis? What statistical methods/tools are going to be useful for me? I've been told to explore outliers, transformations, model selection, model checking and other 'strange behaviours'. Would multiple regression be useful?

This all seems quite overwhelming at the start, could anyone give me a hand pointing me in a direction to begin with? (I am going to be working with R to perform this analysis, too.)

• Multiple regression would seem the way forward but you have many questions about what to do after you have fitted the model which would take a whole book to answer. Software advice is off-topic here but you need to look at lm() and then search this site for the other topics you mention. Plots of the model fit would be a good way to start. – mdewey Nov 24 '20 at 15:08

It appears that your primary concern is conducting the analysis, rather than theoretically exploring the underpinning factors driving house prices. Here is how I would recommend proceeding:

1. Inspect the values of your 15 variables for any missing observations. If you have complete observations for all variables, you are good to go. Otherwise, you might consider if observations in some variables are missing at random, not missing at random, etc. Based on what you find, imputation might need to be conducted, or you will have to run your analysis on fewer observations. Here is a resource for more details.

2. Generate a correlation matrix among all 15 variables (house prices and 14 factors). Examine all pairwise correlation values. High positive or negative correlation between the dependent variable (price) and any of the factors provides descriptive evidence of a significant relationship between the two. You will verify this using regression analysis later. Additionally, if a correlation between two independent variables (i.e. factors) is particularly high (perhaps greater than 0.85), then you might have some collinearity considerations: high overlap in their values and you might not be able to estimate an individual effect of each in a regression). You might want to only use one of the two collinear factors.

3. Generate summary statistics for all 15 variables. Examine the standard deviations, minimum values, and maximum values. This will allow you to see any outliers.

4. Visualize your data with some plots. Use a histogram to determine the distribution of your dependent variable. Scatter plots between the dependent variable (price) and individual independent variables could be interesting. You could also look at how price is contingent upon two variables simultaneously. This will allow you to observe any interesting trends. For example, perhaps the effect of taxes on price is contingent upon levels of air quality. Taxes have a greater positive effect on price when air quality levels are poor, relative to when air quality levels are good. Choose visualizations based on your theory as to why some variables are linked. Another important visualization to generate is the residual plot to test heteroskedasticity (which refers to unequal variable in the dependent variable). Here is are two helpful resource (A and B) for understanding heteroskedasticity and determining its presence using residual plots in R.

5. Run a multiple linear regression predicting price as a function of the final set of factors you have chosen based on the previous steps. (Note: For the time being, I assume you have data for one year and therefore will be conducting a cross-sectional OLS analysis. If you have data for multiple years and want to run more sophisticated panel data methods, here is a helpful resource). If you have heteroskedasticity, you will want to include robust standard errors (shown in previous resources). For each independent variable, carefully examine the coefficients (magnitude, direction: positive/negative), p-values, and confidence intervals. There are many resources available on the internet explaining how to interpret each; here is one. P-values less than 0.1 for particular independent variables generally suggest that that particular factor will have a significant effect on price (at least in your sample). The p-value is a not a strict cutoff, and more of a heuristic. However, for the purposes of your report, the factors that have significant p-values are the ones you will want to focus your efforts on.

6. Building on discussions of the dependent variable being contingent on two independent variables simultaneously, you could also consider interactions in the multiple regression. Many resources available once again, here is one.

Statistical analysis can often be an exploratory or confirmatory endeavor. Much of the steps conducted are based on intuition and heuristics. Your task appears exploratory in nature and I would recommend examining as many descriptive statistics, plots, graphs, and regressions as possible before draining final conclusions. The steps I have discussed above are very rough and quick summaries. There are many more sub-steps you can take within each.

Good luck!

• Incredible answer. Extremely helpful, Anavir. Thank you!! – Mesrick Nov 27 '20 at 10:41

Interesting project to forecast a real estate's selling price (aka, an indicator of value, that is an accepted measure for valuation purposes). However, it is inherently difficult to render a consistent valuation opinion, given this sector inter-relationship (correlation) with harder to forecast economic and other key variables. However, the extent of a lag effect does permit a shorter-term forecast. An example of a significant macro variable includes interest rates, which can/should be decisive in a buyer's decision to rent or buy. The strength of the job market and stock market (impacting 401K, which can finance a down payment) are other examples. Unfortunately, the latter are all subject to somewhat random short term shocks from events including pandemics, natural disasters and even election results (potentially impacting government policies/subsidies),...

And further, there is an apparent somewhat longer-term macro real estate cycle, which can contribute to forecast error. To quote a source:

Real estate markets perform cyclically. The cycles affect output and the absorption of units, and they influence the prices and rents of existing properties and new construction. Expectations for rent increases and the time to start and continue construction are central to the structure of real estate cycles. When participants under forecast rent increases, serially correlated unexpected excess returns trigger construction even if contractors distinguish between relative and nominal price changes. Prices in real estate markets depend on the behavior of the cycle, which in turn affects production and prices. The findings for multifamily housing in Phoenix and Tucson, for example, suggest that cycles are characterized by upside and downside lengths of three years.

Concomitant with the above there is a longer-term demographic shifts (which continue to evolve due to climate change) occurring with older retired people moving to warmer and popular locations both domestically and even possibly foreign countries.

Nevertheless, for the bold, if you have to assess valuation, I would recommend doing so for only a limited horizon.

General model recommendations, a parsimonious model (few variables), as I would employ an auto-regressive time series models with but a few key available variables that have been historically successful in calling turning points at a lag to a market decline.

• It's not clear this project is about forecasting, just modeling current prices. – gung - Reinstate Monica Nov 24 '20 at 14:04
• The 'value' of your home is related to its 'market price'. How does one model 'current prices' absent a 'market price' assessment? Further, to quote from the question: "My goal is to try and fit an appropriate model to the data, which predicts the house price from the other variables". – AJKOER Nov 24 '20 at 14:16
• Thank you for your answer, but indeed this is purely a statistical modelling question. I only want to implement statistical techniques I have studied, such as regression, to analyse the data, not to forecast anything. – Mesrick Nov 24 '20 at 14:17
• Mesrick: A regression model based on a previous sale price of the home (adj to current year), would be an example of my suggested regression model which includes an autoregressive explanatory variable (prior market value of y). Note, if that neighborhood has a bad school, high crime rate, significant pollution problem, flood issue, wildfire exposure, etc, all of that is reflected in the prior selling price for a home of its size, # of bedrooms, etc. You do not need to assess the impact on market price if the home has one of the mentioned conditions, the prior market price has already. – AJKOER Nov 24 '20 at 14:32
• I have re-worded my answer and added sources for those unacquainted with economic valuation models and such. – AJKOER Nov 24 '20 at 17:24