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I am struggling with assessing the best weighted regression type/model specification for my problem.

The goal is to determine influence of independent variables towards the outcome (dependent variable), so estimators and their standard errors should not be biased.

Dependent variable is a proportion (so ranging between 0 and 1), while close to zero means worse outcome of observed phenomenon. However, observations close to 1 are mostly outliers, which behaviour is comparable with lower outcomes (near zero).

Distribution of Dependent variable

When applying weighted linear regression, some of the predicted values are outside the interval for possible outcomes (fitted values less than zero). I assume this is causing an assumption violation of autocorrelation in residuals - see graph 2 (AR tested by Breusch–Pagan test and ACF plot). Can this assumption be right?

fitted values vs residuals

Also searched for different types of regressions to deal with this problem. I tried to apply GLM with family distribution which corresponds with distribution of dependent variable (tried package fitdistrplus in R, but none of distributions available here are close to the outcome variable).

Independent variables have no sigmoidal relationship with the dependent variable, so I assume the GLM with binomial log link and beta regression is not suitable for this problem (Am I wrong??).

Regression needs to be weighted as it comes from significant assumptions of data.

Independent variables are:

  • x1 = categorical with 7 lvls
  • x2 = nominal variable with more than 30 lvls
  • x3 = count of occurrences (logarithmic/linear relationship with dependent variable)
  • x4 = Standardized variable ranging from 0 to 1 ()
  • x5 = continuous variable (growth curve with asymptote)

Can you please help me with assessing the best way to model the data? Or at least point out the way I should be looking at this problem?

EDIT:

To supplement more information regarding the data. I am trying to examine what affects the effectiveness of public procurement contracts (from specific point of view)

This is dataset after removing outliers. They were identified by isolation forest. There was need to take into account different behavioral patterns which are present in the data.

Dependent variable (proportion) = (final price of a contract minus estimated price of a contract)/estimated price of a contract. Describes how effective the procurement was, proportional amount of funds being saved.

Independent variables:

  • (x1) Ordinal values with 7 lvls = Describing different mean values when observation belongs to a given category, expected result is, that with higher ordinal level raises also the dependent variable
  • (x2) Nominal with more than 30 lvls = Describing different mean values when observation belongs to a given category, control variable.
  • (x3) count of occurrences = How many contractors attended the contract
  • (x4) Standardized values = Min Max Standardization of another (fully independent) proportional value on groupby lvl of x2 variable.
  • (x5) continuous variable = describes mean bid count per contractor in a contract. Growth curve with asymptote is derived from the graph below. Describes competitiveness in a contract, which as I expect, has some limitation regarding the effect on a dependent variable.

Below you can see weighted loess on three input variables to see their dependence to the outcome.

X3 vs Dependent variable

X4 vs Dependent variable

X5 vs Dependent variable

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A key principle (but one quite easy to miss) is that as residual $=$ observed MINUS fitted, so each distinct value of the observed defines a straight line with slope $-1$ and intercept that value. So, the residuals from a regression in your set-up must plot in a zone with lower limit

$0$ MINUS fitted

and upper limit

$1$ MINUS fitted

and given the high skewness of your outcome, it should be no surprise that the lower limit is clearly evident as a hard limit, but the upper limit is discernible once you know to look for it. This structure on a residual versus fitted plot is a clear consequence of the bounds of the outcome.

Aside: Is it this that you are calling autocorrelation? Autocorrelation arises from dependence in time or space. The description of your variables is vague, but an allusion to a growth curve hints at some time element. But the structure of the residual versus fitted plot tells us nothing about autocorrelation.

Although the bounds of the outcome are strongly relevant to the kind of model that is suitable, it's a mistake to focus in detail on the marginal distribution of the outcome otherwise, as almost all that matters is to do with the outcome distribution conditional on the predictors.

I would go for a generalized model with logit link and quasi-binomial family with "robust" standard errors, but from your extreme skewness it does seem likely that log link may give similar answers, bearing in mind that

$\text{logit}\ p = \log\ [p / (1 - p)] = \log\ p - \log\ (1 - p)$

is close to $\log\ p$ when $p$ is small, as then $\log (1 - p)$ is close to $\log 1 = 0$.

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  • $\begingroup$ Your insight was very helpful. Thank you. I added more information about variables in the EDIT. I do not expect to be autocorrelation in residuals due to time dimension, but I can't see any other logic reason why the AR is present $\endgroup$
    – Meowcapone
    Commented Nov 13, 2022 at 2:08
  • $\begingroup$ Thanks for the extra information. I can't follow what you are saying about autocorrelation. Perhaps you need a different model structure. $\endgroup$
    – Nick Cox
    Commented Nov 13, 2022 at 8:28

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