For the case of timeseries data first OLS should be estimated so that all possible post regression diagnostics can be done on the bases of present issues appropriate model can be used.
Diagnostic  Source  Model / Solution  
Time Series Data  Non Normal Residuals (JB Test)  Too many outliers (Kurtosis >> 3)  Use Logarithm Model 
Dependent is count variable  Use Poisson Regression  
Multicollinearity (VIF)  Sample Too small  Use single variable regression  
Variables too many  Stepwise regression  
Indirect independent variables  Indexed variables using Principle Factor analysis  
One independent is catalysis for another  Moderator approach  
Few independent variables  Use detrended variables  
Hetroskedasticity (BP Test)  Nonlinear variances in high frequency data  ARCH / GARCH model  
Dependent is binomial dummy  Logit / probit model  
Dependent is multinomial dummy  Multinomial logit model  
MisSpecification  Nonlinear form missing  Incorporate nonlinear form or
Use Logarithm Model 

Instability  Data might have two or more different qualities  Use independent dummy variable for quality change or event (structural break)  
Autocorrelation (DW)  Some quality is interconnecting the time series residuals  ECM / ARDL
require unit root test and cointegration 

Endogeneity  Valid Regression is other way around  Reverse the regression or use IV / GMM regression  
Contemporaneous Correlation / simultaneity  Regression is two way with same other independent variables  VAR / VECM
require unit root test and cointegration 

Two equations are interrelated with same independent variables  VAR / VECM
require unit root test and cointegration 

Two equations are theoretically related with different independent variables.  SURE / SEM 
These models are based on statistical characteristics of the data, there are other models available too which can be used in special case for time series data.
This table provides tentative model selection criterion, they are open for suggestions.
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All the said procedure is to be followed once the unit root test is done or we need anything else before the model selection?
no only the models which are followed by autocorrelation need unit root tests.
Which model do I use when I’m faced with a combination of I(1) and I(2) variables?
Use more than one unit root test to confirm I(2) variable. Then after confirmation take first difference of I(2) variable it will become I(1) variable. Then all variables will become I(1).
Is it proper if y is the first difference term but x is not. Because y is stationary at I (1) only if it is first differenced, otherwise, it is stationary at I(2). X is integrated at I(1).
you can do it, only if the first differenced variable is meaningful. Secondly, I(2) variables are highly rare so confirm using more than one unit root tests.
Different unit root test show different order of integration what should be considered.
discuss the majority decision.
Thank you so much. Your blog helps me a lot.