# Why should we care if the “squares of returns are independently distributed over time” to choose an adequate model of the distribution of returns?

In a Time Series Book by Hashem Pesaran, he mentions that there are a number of issues that need to be addressed in order to choose an adequate model for predicting asset returns.

I understand the other 4 considerations in the picture but I don't understand what it means for the squares or absolute values of returns to be independently distributed over time? Why is that different from the distribution being constant over time which includes the variance being constant over time?

• Squared returns are closely linked to volatility: if mean returns are zero you can see that from the definition of standard deviation. For simple asset models, it is easier to work with the assumption that volatility is stationary. However, this by no means is required.Think of garch like models in econometrics or stochastic volatility models in derivatives pricing for example – Bram Oct 24 '17 at 18:59

First it is not returns but residuals and squared residuals of your model that should be IID.

Second, it is required that squared residuals are IID because you are not only interested on your mean forecast but also about the variance of your forecast.

Imagine two cases :

1. your forecasted value is 0.5 but its variance is unknown because residuals of your models have time-varying variance. The variance of your forecast may be 0.2 or 0.8, you simply don't know it. (in fact you only know its central tendency)
2. your forecasted value is still 0.5 but you know for sure that it has a certain variance (let's say 0.2) because it is derived of the variance of your squared residuals that is constant.

In the second case you are able to build a confidence interval about your forecast but not in the first case.

To sum up, residuals must be IID to be sure that the mean forecast is correct. Squared residuals must be IID to infer the variance of your forecast and to be able to build multi-step forecast and confidence bands.

• The book does refer to returns, not residuals. Given the author I wouldn't think that's a typo. What would that mean in terms of returns? My confusion is that if it refers to the variance of returns isn't that already captured by the point addressing "is the distribution constant over time" since that implies constant variance? – JorgeT Oct 25 '17 at 17:22
• 1) There is no typo, the point is that it is because returns and squared returns are not IID that we model returns with a mean and variance process. At the end if we succeed (squared)residuals should be IID. 2) About your confusion yes, they are the same and to know if the distribution is constant you need to check the squared returns. Ps : your question is misspecified, it should be "Why shoud we ... to choose an adequate model "( and not a good distributions for returns) – Malick Oct 25 '17 at 20:33
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8269868493080139, "perplexity": 452.38053323717304}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999218.7/warc/CC-MAIN-20190620125520-20190620151520-00356.warc.gz"}