The following method works in IE11+, Firefox 25+ and Chrome 30+:
<a id="export" class="myButton" download="" href="#">export</a>
<script>
    function createDownloadLink(anchorSelector, str, fileName){
        if(window.navigator.msSaveOrOpenBlob) {
            var fileData = [str];
            blobObject = new Blob(fileData);
            $(anchorSelector).click(function(){
                window.navigator.msSaveOrOpenBlob(blobObject, fileName);
            });
        } else {
            var url = "data:text/plain;charset=utf-8," + encodeURIComponent(str);
            $(anchorSelector).attr("download", fileName);
            $(anchorSelector).attr("href", url);
        }
    }
    $(function () {
        var str = "hi,file";
        createDownloadLink("#export",str,"file.txt");
    });
</script>
See this in action: JS Fiddle
Firefox and Chrome support data URI for navigation, which allows us to create files by navigating to a data URI, while IE doesn't support it for security purposes. 
On the other hand, IE has API for saving a blob, which can be used to create and download files.